Design of FRPs in Circular Bridge Column Retrofits for Ductility Enhancement

Engineering Structures 30 (2008) 766–776www.elsevier.com/locate/engstruct

Design of FRPs in circular bridge column retrofits for ductility enhancement

Baris Binici∗

Middle East Technical University, Inonu Bulvarı, Department of Civil Engineering, 06531, Ankara, Turkey

Received 10 December 2005; received in revised form 27 January 2007; accepted 10 May 2007Available online 2 July 2007

Abstract

Retrofit design of existing columns in buildings and bridge piers necessitates the accurate prediction of the deformation capacity of structures.In this study, an analytical model is proposed to estimate the ductility of potential plastic hinge regions of RC columns after a fiber reinforcedpolymer (FRP) retrofit. A simple bilinear stress–strain model that is capable of representing an FRP concrete response exhibiting softening atlow confining pressures and hardening at higher confinement is proposed. This model is then employed in an approximate closed form sectionalanalysis of circular columns subjected to axial force and bending moment. Section ductilities that can be obtained as a result of FRP retrofit areexpressed in terms of non-dimensional column parameters and confining pressure supplied by FRPs. The accuracy of the model to estimate theductility of FRP retrofitted columns is verified by comparing model estimations with sectional analysis and test results. Subsequently, a simplenon-iterative seismic retrofit design procedure using FRPs is established for circular bridge columns. Finally, parametric studies are conducted ona typical bridge column for different axial loads, reinforcement ratios and FRP amounts. Results are presented in the form of non-dimensionalplots to aid engineers in the FRP retrofit design of columns for ductility enhancement.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Fiber reinforced polymers; Seismic retrofit; Confinement; Ductility

1. Introduction

Fiber reinforced polymers (FRPs) have become popularin structural retrofit applications due to their appealingadvantages such as being lightweight, of high strength, andeasy to apply. One of the most attractive applications ofFRPs is wrapping existing deficient reinforced concrete bridgecolumns to enhance their deformation capacity, especiallyat the potential plastic hinge regions. In this way, it ispossible to eliminate brittle behavior due to the absence ofproperly designed confining steel reinforcement and to enhancethe deformability of the structural members by avoiding ordelaying undesirable phenomena such as concrete crushing, andlongitudinal reinforcing bar buckling.

There exists a vast amount of experimental work conductedon the FRP retrofit of columns in the last decade.Great emphasis has been placed on strength and ductilityenhancement for concentrically loaded concrete specimensretrofitted with FRPs [1–5]. For FRP wrapped concrete, lateral

∗ Tel.: +90 312 210 2457; fax: +90 312 210 1193.E-mail address: [email protected].

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.05.012

confining pressure is proportional to the axial load up to thepoint where FRP ruptures and failure occur in a sudden andbrittle manner. The increasing nature of confining pressure forFRP wrapped columns generally results in a bilinear responsegiven that the lateral confining pressure is sufficient. Otherstudies have concentrated on the FRP retrofit of deficientRC columns for seismic strengthening [6–9]. It has beenshown that an FRP retrofit can significantly improve the lateraldeformation capacity of columns, keeping the plastic hingeregions intact even at large deformation cycles.

In light of the experimental studies summarized above,a better understanding of the FRP confinement mechanismis achieved and many FRP confined concrete models weredeveloped. An extensive review of the literature on FRPconfined concrete can be found in [10–12]. Most of thesemodels are empirical in nature and employ best fit expressionsas a function of the jacket properties to the experimentallyobtained stress–strain curves. As pointed out by Wu et al. [13],these models are calibrated only for FRP wrapped concretethat exhibits a hardening behavior. However, for large diameterbridge columns it is not always feasible to design for suchconfining pressures. Hence, simple models that are capable

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B. Binici / Engineering Structures 30 (2008) 766–776 767

of representing FRP confined concrete behavior ranging fromsoftening to hardening response for different lateral pressuresare needed. Recently proposed analytical models [14–16]define the axial and lateral stress–strain relations of concretefor different levels of confinement. By matching the expansionof concrete to the straining of the jacket, an FRP confinedconcrete response is obtained from a family of active confinedconcrete curves. These models are analysis oriented and are notsuitable for design. Fewer studies have been conducted on theuse of FRP confined concrete models to estimate the responseof columns subjected to combined axial force and bendingmoment [8,17,18,20]. Some of these models are empirical [8];others are valid only for low levels of confinement [18] and arenot verified by experiment [17,20]. Hence, simple and rationalmodels that can relate the seismic demands directly to FRPdesign parameters (such as the FRP amount) are yet to bedeveloped.

The main objective of this paper is to propose a simplifiedanalytical model to estimate the ductility of an FRP retrofitfor circular columns at section and member levels. In orderto achieve this objective, a simple bilinear stress–strain modelfor FRP confined concrete is employed along with a closedform approximate solution. The developed model can beused directly to estimate the required confining pressurefor estimated seismic ductility demand on the structure.The accuracy of the model is verified by comparing modelestimations with numerical sectional analysis and test results. Itis believed that the modeling and design approach proposed inthis study will provide structural engineers with a clear, simpleand systematic way of evaluating and retrofitting deficientcircular columns.

2. FRP confined concrete model

2.1. Stress–strain behavior

Experiments conducted on FRP confined concrete cylindersand prisms reveal that axial compressive stress–strain behaviorof FRP confined concrete exhibits almost a bilinear response.A number of nonlinear models were proposed in previousstudies to describe the complete stress–strain response ofFRP confined concrete [12,19]. In a recent study by Saiidiet al. [20], it was shown that a bilinear idealization for thestress–strain response of FRP confined concrete can yieldaccurate estimates of strength and deformation capacity ofcolumns when compared to estimates obtained through theuse of more complicated material models. Hence, a simplebilinear stress–strain relationship for the compressive behaviorof FRP confined concrete is adapted in this study withoutsignificant loss of accuracy. The simple model presented belowcovers the stress–strain response of confined concrete from lowconfinement showing softening behavior to high confinementwith hardening behavior.

The strength and ductility of FRP confined concrete dependson the amount of lateral confining pressure. The confinementratio, φ, is given as:

φ =E f ε f n f t f

f ′c R

(1)

Fig. 1. FRP confined concrete model and rectangular stress block parameters.

Fig. 2. Comparisons of strength estimations with experimental results.

where E f and ε f are the modulus of elasticity and rupturestrain of the FRP jacket in the hoop direction, t f is the thicknessof one layer FRP jacket, n f is the number of FRP layers, Ris the radius of the confined concrete section, and f ′

c is theunconfined concrete compressive strength. When φ is smallerthan a certain transition value (φt ≈ 0.1–0.15), the stress–strainresponse was observed to exhibit a softening region. Samaanet al. [4] argued that the transition value is 0.15 whereas Wuet al. [13] reported this value to be 0.13 based on a largedatabase of test results. An average value of 0.14 is used in thisstudy as the transition value.

Two stress–strain pairs, namely the breakpoint and theultimate point, are sufficient to define the complete stress–strainresponse of FRP confined concrete (Fig. 1). The breakpoint ofthe stress–strain curve is assumed to occur at point (εo, f ′

c)(where εo is usually assumed to be 0.002, similar to strain atpeak stress for unconfined concrete) up to which the behavioris elastic. It has been pointed out previously that the deviationof this breakpoint from the assumed value in this study isnegligible [12]. The ultimate strength enhancement factor, Kσ ,is given as a function of φ and φt as:

Kσ =

{2.6(φ − 0.14)0.17

+ 1 φ ≥ φt

1.8φ0.3 φ ≤ φt .(2)

Comparisons of strength estimations using Eq. (2) and testresults from Refs. [4,5,12] are shown in Fig. 2. It can beobserved that the ultimate strength for both softening andhardening specimens can be well represented with the proposedequations. In addition, Eq. (2) is more convenient to use

768 B. Binici / Engineering Structures 30 (2008) 766–776

(a) Ref. [5]. (b) Ref. [2].

Fig. 3. Comparisons of axial response estimations with experimental results (dots are experimental points, lines are model estimations).

compared to strength envelopes proposed in other studies [4,10] as it provides a smooth transition from the softening tohardening response for different confinement ratios. The strainenhancement factor, Kε, as a result of confinement is taken asproposed by Lam and Teng [12] based on a calibration of 76FRP confined concrete specimens exhibiting both softening andhardening responses:

Kε = 1.75 + 12φε (3)

where ε = (ε f /εo)0.45. Axial response estimations using

the simple bilinear model are presented in Fig. 3 along withresults from two sets of carbon fiber reinforced polymer (CFRP)confined concrete tests [2,5]. Test variables were the ratio ofsquare specimen size to corner radius in tests conducted byRochette and Labosiere [5] and the number of CFRP pliesin tests by Harries and Kharel [2]. In order to compute φfor specimens in [5], a confinement effectiveness factor asproposed by Eurocode8 [21] was used. It can be observed thatthe model estimations of strength and deformation capacitiesof test specimens are within ±15% of test results. Furthermorethe softening hardening transition is estimated with sufficientengineering accuracy.

2.2. Rectangular stress block parameters

In order to make use of the simple bilinear stress–strainmodel for FRP confined concrete explained above in asectional analysis for the ultimate state, rectangular stressblock parameters α and β are computed (Fig. 1). Parameterβ is computed such that the centroid of the rectangular stressblock and the area under the stress–strain curve coincide. Thenparameter α can be computed from the equivalency of therectangular stress block and the area under the stress–straincurve. Calculated values from this equivalency for α and β areplotted as a function of φ in Fig. 4. It should be noted that thesevalues are not sensitive to the selection of ε within the practicalrange (ε ranging from 4 to 12). As a result of a linear regressionanalysis, α can be approximated as:

α = 0.85 + 1.64φ. (4)

On the other hand β is merely sensitive to the confinement leveland can be taken as 0.85 for all practical purposes (Fig. 4).

Fig. 4. Computed and approximate rectangular stress block parameters.

3. Ductility of FRP retrofitted circular columns

In this section, ductility of a circular bridge column confinedwith FRPs and subjected to a combined axial load and bendingmoment is examined. With the objective of obtaining anexplicit relationship between section curvature ductility andnormalized confinement ratio, φ, a number of simplifyingassumptions were made. The classical Bernoulli–Euler beamtheory is adopted where plane sections remain plane prior toand after deformations. Perfect bond is assumed between thelongitudinal steel reinforcement and concrete and between theFRP jacket and concrete. The concrete tensile load carryingcapacity is neglected and steel reinforcement is assumed asan elastic perfectly plastic material. The shear capacity of theretrofitted column is assumed to be higher than the capacitydetermined by flexural strength. It is further assumed thatthe longitudinal steel reinforcement is continuous along themember’s axis and distributed uniformly around a circularperimeter. In order words, longitudinal steel is smeared arounda circular perimeter. This assumption is believed to be realisticfor bridge columns where a large number of bars are used forlongitudinal reinforcement. Furthermore, concrete stresses inthe compression zone are assumed to be well represented by arectangular stress block. It should be mentioned that concretein the compression zone is assumed to be described by thebilinear FRP confined concrete model described above. Dueto the strain gradient in the cross-section, it can be arguedthat concrete fibers parallel to the neutral axis experience


Fig. 5. Sectional analysis of a circular column retrofitted with FRPs.

different levels of confining stress in the compression zone.This situation is valid both for FRP confinement and steelconfined concrete prior to yielding of transverse steel. However,it is reasonable to accept that for fibers close to the neutralaxis, axial strain level is relatively low such that the effectof confinement on the axial stress is negligible. For fibersfurther away from the neutral axis the bilinear stress–strainmodel can be expected to represent the stress–strain responsewith sufficient accuracy. A similar assumption was previouslyemployed by many researchers for the sectional analysis ofsteel confined [22–24] and FRP confined [9,18,20] reinforcedconcrete sections. More advanced models that take into accountthe confining stress variation in the compression zone werealso proposed [25,26] for use along with fiber frame finiteelements. These models are very sophisticated and requireiterative procedures and immense post-processing. Hence theyare not currently appropriate for design and their use in thecurrent study is not further elaborated and attention is focusedon a more simplified method of sectional analysis.

Following the derivations given in the next section,comparisons of the model predictions with test results andvalidity of the above assumptions are examined using numericalanalysis and the available test results.

3.1. Section ductility

Consider a circular column with a radius of R, subjected tocombined axial load, Pa , and bending moment (Fig. 5). First,the analysis of the section at the ultimate state, which is definedby the rupture of the FRP jacket (i.e. when the extreme fiber incompression reaches a strain of εcu = εo Kε) is presented. Theforce equilibrium of the section can be expressed as:

α f ′c(Acc − A−

s )+ A−s fy − A+

s fy = Pa (5)

where Acc is the area of the compression zone, A−s and A+

sare the areas of longitudinal steel in the compression andtension zones respectively, and fy is the yield strength oflongitudinal reinforcement. It should be noted that both steelin the tension and compression zones are assumed to yieldin Eq. (5). This assumption was employed previously [18]and was justified due to the fact that the contribution of barsclose to the neutral axis cancel out in equilibrium and donot significantly affect the results. Furthermore, since the FRP

Fig. 6. Approximations for trigonometric functions.

retrofitted column can reach εcu values above three times εo,it is reasonable to assume steel yielding at the centroid ofthe smeared longitudinal reinforcement for neutral axis depthsranging from 0.1 to 1.5 times the column radius. Next, concreteand steel areas are expressed as a function of θ and θ ′ (Fig. 5):Acc = 0.5R2(θ − sin θ), A−

s = ρs Ag(θ′/2π) and A+

s =

ρs Ag(1 − θ ′/2π) (where Ag = πR2). Recognizing the factthat concrete cover is small compared to the radius for bridgecolumns, θ ≈ θ ′ can be assumed. Finally, neglecting A−

s in thefirst term without loss of accuracy and dividing both sides byf ′c Ag , we obtain:

α

2π(θ − sin θ)+ I

(θ

π− 1

)= n. (6)

Above, I is named as the reinforcement index in this study (I =

ρs fy/ f ′c), and n is the axial force ratio (n = Pa/πR2 f ′

c). OnceEq. (6) is solved for θ , the corresponding ultimate curvature,ψu , can be found from:

ψu =εcu

c=

βεo KεR(1 − cos(θ/2))

. (7)

It can be observed that trigonometric functions in Eqs. (6)and (7) render it impossible to find a closed-form solutionfor ultimate curvature and curvature ductility. Hence, a linearapproximation is employed within the practical range of interestof θ for the terms (θ − sin θ ) and cos(θ/2) as shown in Fig. 6.The following approximations can be made for these functions


without significant loss of accuracy:

θ − sin(θ) ≈ 1.40θ − 1.54 (8)

cos(θ/2) ≈ 1.38 − 0.44θ. (9)

Note that the above equations result in at most a ±20% error forθ from 1.3 to 6.2 rad (75◦–360◦), which covers almost all thepractical range of neutral axis locations (0.1 to 1 times sectiondiameter) due to combined axial loads and bending moments.The impact of this approximation in estimating the ultimatecurvature is examined in the next section. Combining Eqs. (8)and (6), the angle θ that defines the depth of compression stressblock is:

θ =n + I + 0.4φ + 0.21

0.32I + 0.37φ + 0.19. (10)

Substituting Eqs. (9) and (10) into Eq. (7), and substituting 0.85and 0.002 for β and εo, respectively, the ultimate curvature ofan FRP retrofitted circular section can be obtained from:

ψu =0.0017F(I, n, ε, φ)

R(11a)

F(I, n, ε, φ)

=4.44εφ2

+ (0.65 + 3.84ε(I + 0.59))φ + 0.56I + 0.330.44n + 0.32I + 0.04φ + 0.02

.

(11b)

It can be observed that ultimate curvature is a functionof column radius, (R), normalized FRP rupture strain (ε),reinforcement index (I ), axial load ratio (n), and confinementratio, (φ). In order to estimate the available curvature ductility,yield curvature, ψy , needs to be estimated. For this purpose,a simple relationship, which was shown to have an accuracyof ±15% (yield curvature can vary at most 15% from thatestimated by Eq. (12)), is employed [22]:

ψy =1.225εy

R(12)

where εy is the yield strain of longitudinal bars. It may beargued that the error in the estimated yield curvature usingEq. (12) can result in error for the curvature ductility inaddition to that coming from the error in the ultimate curvatureestimation (∼12% as explained in the next section). However,it should be remembered that the definition of the yield pointis a subtle issue and considering various definitions of it(i.e. steel yielding, concrete compressive strain reaching 0.002or based on a bilinear fit of the moment-curvature response), thetheoretical yield point can show shifts with a similar order ofmagnitude to the approximation error. In addition, it should beremembered that the uncertainties associated with the demand(i.e. earthquake imposed deformations) and capacity (in-situstrength and deformation capacity of materials, constructioninduced errors) can play a much more important role than thoseerrors introduced in Eq. (12). The objective of the present studyis to provide structural engineers with a tool for the quickpreliminary design of FRPs in column retrofits. Hence, it isbelieved that errors in the order of 15% for the yield curvatureand 12% for the ultimate curvature are within engineering

accuracy and can be tolerated, bearing in mind the complexityof the actual problem.

The experimental studies conducted on the behavior ofFRP confined concrete [1–5] revealed that confinement issignificantly active beyond f ′

c , which corresponds to a strainlevel of about 0.002 for normal strength concrete. In fact,a number of proposed FRP confined concrete stress–strainmodels that have been verified with experiments [11] acceptsimilar responses for FRP confined concrete and unconfinedconcrete up to this stress–strain point. Hence, the effect ofconfinement at axial strains smaller than about 0.002 can beconsidered to be negligible. Assuming that for the yield pointgiven by Eq. (12), concrete extreme fiber compressive strainusually corresponds to values less than the uniaxial compressivestrength of unconfined concrete (i.e. under reinforced cases), itis reasonable to neglect the effect of FRP confinement on theyield point. Hence, it can be stated that the presence of FRPsshifts the yield curvature insignificantly. Combining Eqs. (11)and (12), curvature ductility, µψ , of the section can be obtainedby using:

µψ =0.0014εy

F(I, n, ε, φ). (13)

From a design perspective it is necessary to calculate φ, fora required curvature ductility µψr as demanded on the structureby earthquake and vertical loads. This can be accomplished bysolving Eq. (13) for the required normalized confinement ratio,φr , to design the FRP amount. Taking εy as 0.0021 for steelreinforcement having a yield strength of 420 MPa, φr , is:

φr =

(−B +

√B2 − 4AC

)/(2A) (14)

where

A = 4.44ε (15)

B = ε(2.28 + 3.84I )− 0.06µψr + 0.65 (16)

C = 0.56I + 0.33 − 1.5µψr (0.44n + 0.32I + 0.02). (17)

In this way, it is possible to design FRPs based on an estimatedcurvature ductility demand without having the need to conductany iterative numerical analysis. This method is extended toa displacement ductility based design approach in the nextsection.

Finally, moment capacity of the section is computedassuming the tension steel to be located at the centroid ofthe distributed reinforcing layer. Summing the moment offorces about the concrete compressive force and neglectingthe contribution from the compression reinforcement, momentcapacity, Mu of the section is:

Mu = f ′cπR3G(I, n, θ) (18a)

G(I, n, θ) =

{I

(1 −

θ

2π

) (0.5 +

cos(θ/2)2

+sin(π − θ/2)

θ/2

)+ n

(0.5 +

cos(θ/2)2

)}. (18b)

Above, moment capacity is described as a function of θ , whichis in turn a function of confinement ratio Eq. (10). Hence,


Fig. 7. Normalized interaction diagrams for FRP wrapped circular columns(I = 0.2).

utilizing Eqs. (10) and (18) it is possible to obtain normalizedinteraction diagrams for different levels of confinement. Anexample of such a diagram is given for confinement ratios of0.1–0.4 and for I equal to 0.2 in Fig. 7. It can be observedthat for axial load levels below the balanced point of theunconfined column, moment capacity of the section retrofittedcolumns are affected only up to about 25% for confinementratios below 0.3. This phenomenon was previously notedin experimental and analytical studies [6,8,18]. Therefore,assuming zero confinement to estimate the moment capacityof the section is conservative at this range. In addition,this assumption can introduce great flexibility from a designperspective as it removes the need to perform any iterationwhile computing the moment capacity of the section as afunction of confinement prior to designing the FRP amount.A simplified seismic design procedure employing the abovemodel will be presented following model verification.

3.2. Model verification

First, the analytical model developed to estimate the ultimatecurvature and moment capacity is verified by comparing theresults obtained through the use of a filament based sectionalanalysis. The bilinear constitutive model for compression isadopted in the sectional analysis assuming no tensile strengthof concrete. An elastic-perfectly plastic with strain hardeningmodel for steel reinforcement is employed. A typical bridgepier having a diameter of one meter and clear cover of40 mm with a compressive strength of 30 MPa is selectedfor the analysis. The longitudinal reinforcement ratio andaxial load ratio are varied between 0.01%–0.04% and 0.1–0.4with 0.01 and 0.1 increments, respectively. Note that asimilar range of parameters was previously investigated todetermine the performance levels of bridge columns withcode compliant transverse reinforcement designs [22]. Carbonreinforced polymers with a modulus of elasticity of 75 000MPa and a rupture strain of 0.015 was employed assumingthat existing transverse steel provides a negligible amount ofconfinement. The number of CFRP layers (having a thicknessof 1 mm per layer) was varied from one to five with one-layerincrements. The ultimate curvature and moment capacity ofthe retrofitted columns are compared with model estimations

obtained through the use of Eqs. (11) and (18), respectively(Fig. 8). Note that up to the ultimate curvature defined byrupture of the FRP sheets, moment curvature response doesexhibit a descending branch. An excellent agreement can beobserved between ultimate curvatures obtained using Eq. (11)and numerical sectional analysis, while the computation cost ofthe analytical model is minimal when compared to numericalanalysis. In the previous section, it was mentioned that dueto the use of simple approximate equations given by Eqs. (8)and (9), the highest error committed is in the order of 20%.It can be observed from Fig. 8 that the mean of the ratios ofultimate curvatures estimated from sectional analysis to thosefrom the simplified method is 0.98 with a standard deviation of0.05. This clearly shows that the approximations used in Eqs.(8) and (9) (at most 20%) do not accumulate and in fact theerror in the ultimate curvature values (at most 12%) is much lessthan those observed in approximate functions employed in thederivation of the model. Furthermore, Eq. (11) provides a clearvision of important parameters affecting the ultimate curvatureof the section. The moment capacity estimations obtainedthrough Eq. (18) assuming an FRP confined concrete modeland unconfined concrete model are presented in Fig. 8. It canbe observed that FRP confinement has a negligible influenceon moment capacity estimations. All the approximate momentcapacity estimations are within 90% of moment capacitiesobtained using numerical sectional analysis. These verificationsprovide confidence on the validity of numerical approximationsmade while developing Eqs. (11) and (18). The advantage ofthe model developed in this study is apparent as it employsexplicit expressions relating ultimate curvatures to confinementprovided by the FRPs.

An approximate sectional analysis model developed above isalso verified by comparing experimental results reported in [6].It should be noted that to the knowledge of the author, this is theonly experimental study on FRP wrapped circular reinforcedconcrete columns that reported moment-curvature results inthe plastic hinge region. An experimental program [6] wasconducted to observe the effects of two types of FRP lamina,namely carbon (CFRP) and glass (GFRP), on the behaviorof retrofitted building columns with confining transverse steeldeficiency subjected to constant axial loads. Two different axialload levels (n ≈ 0.3 and 0.6) and two different FRP lamina(CFRP and GFRP) were taken as the test parameters for circularcolumns having a diameter of 356 mm with a clear cover of21 mm. All specimens had a uniaxial concrete compressivestrength of about 40 MPa whereas the yield strength oflongitudinal steel reinforcement was about 495 MPa. Moment-curvature response obtained from LVDT’s located within theplastic hinge region were reported for all the specimens. Theexperimental and analytical moment-curvature responses forFRP-retrofitted specimens are presented in Fig. 9 along withmaterial properties of FRPs obtained from coupon tests. It canbe observed that the strength, deformation capacity and FRPrupture deformation levels are in good agreement for specimensST-2NT, ST-3NT with high axial loads. On the other hand, thedeformation capacity of specimen ST-4NT is underestimatedby about 20%. Although the strength of specimen ST5-NT was


Fig. 8. Comparisons of sectional analysis results with model estimations for ultimate curvature and moment capacity.

Fig. 9. Comparisons of experimental [6] and analytical moment curvature responses (solid lines are analytical, dashed lines are experimental).

estimated with reasonable accuracy, the deformation capacitywas significantly underestimated. GFRP wrapping in specimenST-5NT resulted in about half the confinement provided inspecimen in ST-4NT by CFRP wrapping. However, the ductilityof the specimen in ST-5NT is higher than that of specimen ST-4NT in the experiments. This discrepancy in the experimentalresults of specimen ST-5NT contradicts engineering intuition.The researchers of this experimental study did not elaborateon the existing discrepancy of their test results (i.e. achievingsimilar ultimate curvatures with different confinement levels)as they did not conduct any qualitative study on the obtainedexperimental results. It can be argued that the in-situ strengthand deformation capacity of GFRP fabric can vary due tothe uncontrolled impregnation process used by the researchers.This, in addition to other experimental bias, may have resultedin such an unexpected response. Since the objective of thisstudy is to verify the proposed simple model with availableexperimental moment-curvature results, further discussion ofexperimental results is deemed unnecessary here. However, itcan be stated that except for this specimen with unexpectedly

large deformation capacity, analytical estimations of simplifiedmoment curvature relations can be accepted to have sufficientaccuracy when compared to available experimental results.

3.3. Member ductility and seismic design

The lateral displacement capacity of a circular column fixedat its base and free at the top can be related to the sectionalcurvatures through the use of the well-established plastic hingeconcept [22]. Accordingly, lateral top displacement, ∆u , can befound from:

∆u = ∆y + (ψu − ψy)L p(L − 0.5L p) (19)

where ∆y is the lateral displacement at the yielding of the steelreinforcement (∆y = ψy L2/3), L p is the length of the plastichinge along which the curvature is assumed to be uniformlyconcentrated, and L is the shear span. The correspondingdisplacement ductility, µ∆, can be written as:

µ∆ = 1 + 3(µψ − 1)(

1 − 0.5L p

L

)L p

L. (20)


Conversely, the required curvature ductility, µψr , for a givendisplacement ductility demand, µd , is:

µψr =µd − 1

3(1 − 0.5L p/L)(L p/L)+ 1. (21)

Furthermore, the ultimate drift ratio of the bridge column canbe obtained by dividing Eq. (19) by L . Substituting Eqs. (11)and (12) into Eq. (19) and dividing by L , the ultimate drift ratioDRmax is:

DRmax = 0.41εyL

R+ (0.0017F(I, n, ε, φ)− 1.225εy)

×

(L p

L

) (L

R

) (1 − 0.5

L p

L

). (22)

It is interesting to observe that DRmax is expressed, among othernon-dimensional parameters, I, n, ε, only as a function of thecolumn aspect ratio (L/R), the plastic hinge length ratio, L p

L ,and the normalized confining stress, φ.

The bridge piers can be idealized as single degree of freedomstructures with the first mode being the dominant mode ofvibration [22]. Subsequently, it is possible to obtain the ductilitydemand for a specified suite of ground motions by conductinga number of nonlinear time history analyses (NLTHA). Onthe other hand, it is also possible to utilize readily availablerelations for an inelastic design spectrum. The most well-known one, Newmark–Hall [27] constant ductility spectrum, isemployed in this study. According to Newmark and Hall [27],the equal displacement rule can be applied in the constantvelocity region for periods ranging from about 0.5 to 8 s. Inthis region the force reduction factor Ry can be assumed to beequal to the displacement ductility. For a typical free abutmentbridge column with an L/R ratio greater than about 7, it canbe shown that the period of the structure is usually greater thanabout 0.75 s. Hence it is reasonable to assume that Ry , whichcan be computed using Eq. (23), is approximately equal to theductility demand on the structure.

Ry =Sa Pa

Fy=

San

G(I, n, θ)

L

R(23)

where Sa is the spectral acceleration coefficient and Fy is thelateral force to cause first yielding at the column base and iscomputed by Fy = My/L neglecting a second order momentdue to axial force. While obtaining the second equality, My isfound using Eq. (18) assuming zero confinement. In this way,the yield moment of the section is assumed to be equal to themoment capacity of the section prior to any FRP retrofit. Thisassumption is conservative since moment capacity will alwaysbe underestimated resulting in a larger ductility demand.

All the essentials for ductility based seismic designprocedure are laid out in the preceding sections. Hence it ispossible to perform a non-iterative FRP retrofit design for agiven seismic demand as shown with a flow chart in Fig. 10.First, existing column data needs to be collected, includingthe estimations of the reinforcement ratio, concrete strength,axial load, and yield strength of the longitudinal reinforcement.Then, for a suite of site specific ground motion, nonlinear

Fig. 10. FRP retrofit design procedure for bridge columns.

time history analysis can be conducted to estimate the ductilitydemand. Alternatively, assuming the constant velocity regionfor the inelastic design spectrum, a force reduction factorcomputed by Eq. (23) can be accepted to be the same as thedisplacement ductility demand. Then, for the estimated plastichinge length, curvature ductility demand can be found usingEq. (21). Finally, for selected FRP type and material properties,a confinement ratio computed using Eq. (14) can be used todetermine the required number of FRP layers. It should benoted that since estimating the yield moment assuming noconfinement is accurate and conservative, the procedure doesnot involve any iterations. The results of parametric studies andthe outcome of the procedure for common column parametersare presented in the next section.

3.4. Parametric studies

A prototype bridge column with an L/R ratio of 10is selected for the parametric studies. It is assumed thatlongitudinal steel has a yield strain of 0.0021, whichcorresponds to a yield strength of about 420 MPa. The axialload ratio, n, was selected to vary between 0.1 and 0.4 whereasthe longitudinal reinforcement ratio values are assumed to bebetween 0.01 and 0.04. The considered variations in n and I(∼0.1–0.4) are believed to present a broad range of possiblecolumn properties. Two types of fiber reinforced polymers,namely glass (GFRPs) and carbon (CFRP), are selected withan ultimate rupture strain of 0.013 and 0.02, and a modulus ofelasticity of 71 000 MPa, and 21 000 MPa with a ply thicknessof 1 and 1.25 mm, respectively (similar to those used inRef. [6]). Assuming the longitudinal bar diameter, db, to be32 mm, and column length to be 10 m, the normalized plastichinge length is calculated to be about 0.1L by [28]:

L p

L= 0.077 + 8.16

db

L. (24)

The expression above was shown to correlate well with theexperimentally measured plastic hinge lengths of 59 columns


Fig. 11. Curvature ductility–confinement ratio relationships (I = 0.2).

tested under constant axial load and cyclic lateral displacementexcursions [28].

First, the curvature ductility of an FRP retrofitted columnfor a given confinement ratio is computed using Eq. (11)and the results are presented in Fig. 11. It can be observedthat similar curvature ductilities can be attained for increasingaxia1 load ratios by increasing the confinement ratio. Moreover,a confinement ratio of about 0.5 is necessary to obtain acurvature ductility of about 20 for all possible values of nstudied. For similar axial load levels, GFRP retrofit tends toprovide higher curvature ductility as long as the same amountof lateral confining pressure is provided. Note that this doesnot necessarily imply the use of the same number of CFRP andGFRP layers, since GFRPs generally have a lower modulus ofelasticity. This argument can be justified by examining Eqs. (1)and (3). By using the values given in the previous paragraphfor modulus of elasticity, rupture strain and ply thickness forCFRP and GFRP in Eq. (1), it is apparent that to have the sameamount of confinement ratio (φ), approximately 75% moreGFRP layers needs to be used compared to CFRP. In addition,the strain enhancement factor given by Eq. (3) is linearlyproportional to φ and ε0.45

f . Even though the confinementratios are similar, since GFRPs generally have a higher ε f ,the ultimate fiber strain for concrete in compression will behigher for GFRP confined concrete compared to CFRP confinedconcrete resulting in higher ductilities. For the CFRP retrofittedcolumn having an axial load ratio of 0.2, the effect of parameterI on curvature ductility is presented in Fig. 12. Results showthat for the same confinement ratio, curvature ductility islarger for columns with smaller I values. However, curvatureductilities are not very sensitive to values of I for confinementlevels below 0.5.

The maximum attainable drift ratios for the retrofittedcolumns are obtained using Eq. (22) and results are presented inFig. 13. It can be observed that a confinement ratio of about 0.4and 0.5 ensures that the columns can survive drift ratios of about3 and 4%, respectively, irrespective of the axial load level. Fora column having an axial load ratio of 0.2, the drift capacityis almost doubled by only providing a confinement ratio ofabout 0.15. This in fact shows that FRP application is a feasible

Fig. 12. Curvature ductility–confinement ratio relationships (n = 0.2).

Fig. 13. Maximum drift ratio–confinement ratio relationships (I = 0.2).

alternative to consider in column retrofits. The GFRP retrofitresults in a larger drift ratio capacity compared to its CFRPretrofitted counterpart, as long as similar lateral confinementis provided. It is important to mention that other factors suchas durability and fire protection also need to be consideredin the selection of FRP type. The effects of I on maximumattainable drift ratios are presented in Fig. 14 for the CFRPretrofitted column having an axial load ratio of 0.2. As I valuesincrease it is clear that drift ratio capacity decreases for a givenconfinement ratio. The maximum drift ratio correspondingto the collapse prevention limit (beyond which second ordereffects amplify and columns can loose their axial load carryingcapacity) can safely be taken as about 2.5% [29]. From Figs. 13and 14, it can be deduced that a required confinement ratiovaries between 0.1 and 0.3 for this range. Hence, accurateestimations of ultimate concrete strains at confinement levelsbelow the transition values are as important as those at higherconfinement levels for FRP applications to enhance the ductilityof circular columns.


Fig. 14. Maximum drift ratio–confinement ratio relationships (n = 0.2).

Fig. 15. Design confinement ratio for a given spectral acceleration coefficient(I = 0.2).

Finally, required confinement ratios are computed fora specified spectral acceleration coefficient and results arepresented in Fig. 15 for I of 0.2. Note that Eq. (23) is usedto compute the required displacement ductility (µd = Ry),then by using Eqs. (21) and (14) required confinement ratiosare determined. In other words, the design flowchart shown inFig. 9 is used for the column parameters under investigation.Results shown in Fig. 15 emphasize once more that the axialload ratio is a very important factor affecting the designconfinement ratio for a given spectral acceleration coefficient.For a confinement ratio of about 0.3, a typical bridge columnhaving an axial load ratio of 0.1 can survive an earthquakehaving an Sa up to 1. The novel feature of the analytical modeldeveloped in this study is apparent in Fig. 15 and that is itsability to relate a demand parameter (Sa) to a design parameter(φr ) that involves material properties of the retrofit material.Hence a direct relationship between earthquake demand andFRP amount can be established in an approximate manner usingthe modeling approach proposed in this study.

4. Conclusions

A simple analytical model to estimate ductility anddeformation capacity of FRP retrofitted circular bridge columnsis developed in this study. Employing a simple bilinearstress–strain relationship for FRP confined concrete, the modelcan be used directly to estimate the required confining pressurefor given column parameters. The accuracy of the model toestimate the ductility of FRP retrofitted columns is verifiedby comparing model estimations with numerical analysis andtest results. Parameters such as reinforcement index, axialload ratio, and jacket rupture strain are found to influence theexpected ductility of FRP retrofitted columns. The non-iterativeseismic design procedure established in this study provides asimple means of evaluating and retrofitting deficient circularcolumns with FRPs. The following observations can be made,based on the conducted parametric studies: (1) A confinementratio of about 0.4 ensures a curvature ductility of about 30for typical bridge columns having axial load ratios of 0.1–0.2.(2) Displacement ductility and drift ratios of FRP retrofittedcolumns are more sensitive to the axial load ratios than thereinforcement index for the range of parameters studied. (3)GFRPs with higher strain limits provide higher ductilities atsimilar confinement levels at the expense of more layers dueto their lower stiffness. (4) The retrofitted columns with lowaxial loads can survive a drift ratio of 4% for a confinementratio range of about 0.2–0.3.

It is worth mentioning that the proposed modeling approachin this study is approximate. Certainly, further improvementscan be made using more complicated and iterative methodssuch as sectional analysis or fiber based frame finite elements.However, the improvement that would be obtained in estimatingthe actual response of an existing column is uncertain.Therefore, it is believed that the proposed model in this studycan be accepted as a simple tool that can aid the engineersin the assessment and design of FRP retrofits of circularcolumns. Furthermore, it can be used within a probabilisticframework (i.e. first order second moment methods) to establishperformance based design procedures for FRP retrofits as itemploys explicit equations that relate material properties andengineering demand parameters.

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Documents

Design of FRPs in Circular Bridge Column Retrofits for Ductility Enhancement