Deformation analysis of reinforced concrete columns after repair with FRP jacketing S.P. Tastani Democritus University of Thrace, Greece G.E. Thermou Aristotle University of Thessaloniki, Greece S.J. Pantazopoulou U.C.Y. (Cyprus), on leave from D.U.Th. (Greece)

SUMMARY Retrofits of initially damaged brittle r.c. columns through FRP jacketing of plastic hinge zones fall within two categories when considering the damage attained during the pre-repair phase; the first includes specimens that attained a premature mode of failure without developing any ductility and the second one refers to columns that underwent inelastic strains at the critical section. In the latter case the contribution of FRP jacketing is less efficient in terms of strain development capacity of reinforcement, owing to yield penetration into the anchorage which destroys interfacial bond. The strain development capacity of the jacketed anchorage is evaluated by closed form solutions of the field equations of bond over the embedded length considering bond plastification, cover debonding after bar yielding and spread of inelasticity in the anchorage. It is shown that the accumulated damage in the anchorage of such elements reduces post-repair ductility and inelastic deformation capacity even after jacketing. Keywords: assessment, seismic upgrading, FRP jacketing, bond, yield penetration. 1. INTRODUCTION Seismic retrofits that depend on FRP-jacketing for confinement of plastic hinge zones ride on the spectacular improvements reported in terms of strength and deformation capacity of lightly-reinforced columns tested in the lab after strengthening. However the number of tests conducted on repaired (i.e., jacketed after some extent of initial damage had been induced), rather than strengthened specimens through FRP jacketing, is rather limited. In the case of the FRP repaired specimens the level of damage attained during the first loading appears to be critical for the post repair structural response and the associated FRP effectiveness. Irreversible damage of the bar-concrete interface and the associated loss of bond strength owing to extensive yield penetration into the anchorage during initial loading seems to limit the FRP-jacketing effectiveness as compared with the case where the element had attained an initial premature mode of failure, usually by shear, or by lap-splice splitting (Thermou and Pantazopoulou 2009, Syntzirma and Pantazopoulou 2006, Syntzirma et al. 2006). In the FRP-repaired tests some generic observations are that damage is effectively suppressed in the wrapped region at the expense of increased contribution owing to pullout either from the reinforcement anchorage in the footing or from the lap-splice developed in the plastic hinge zone. The fraction of drift capacity due to pullout is greater when reinforcement has undergone large inelastic strains at the critical section during the pre-repair phase of loading. Actually, the increase of the inelastic strain at the critical section has a dually negative impact on the deformability of the post-repair response even if the interfacial bond is fully recovered prior to jacketing; firstly the available inelastic strain decreases by the amount of residual strain consumed in the initial phase of loading, with a consequent result being the limitation of the dependable post-repair flexural drift capacity. Furthermore, if no interventions are implemented within the anchorage, the available anchorage length is shortened by an amount equal to the yield penetration segment which corresponds to a reduction in the effective development capacity. Note that for an anchorage to be fully retrofitted, before the application of the FRP jackets on the member length, it would be necessary to replace the concrete

cover with a new cementitious grout combined with epoxy injection wherever it is suspected that even fine cracks might have been formed in the concrete mass. Comparing the pre- and post-repair resistance curve of the member in terms of load versus lateral drift, it is noted that the initial stiffness of the repaired column is significantly lower than the slope of the ascending branch of a lightly reinforced column having the same details but loaded in pristine condition (stiffness is defined by the secant slope to the point of initial yielding θy

init, or apparent yielding θyR, Fig. 1a). Actually, the

resistance curve of a repaired column that had initially developed some ductility, often presents a concave rather than an arched shape, due to the initial travel of the bar required in order to lock the ribs and mobilize the anchorage. This phenomenon is attributed to the plastic strains experienced by the longitudinal reinforcement during the first loading phase (εs,pl

init) inside the anchorage, which results in a residual inelastic strain εs

res after complete unloading. (From the stress – strain steel law depicted in Fig. 1b, it follows that εs

res=εs,plinit –

Esh/Es(εsoinit-εsy).) This inelastic strain εs

res integrated over the yield penetration length ℓr

init (see Fig. 2) in the anchorage produces a significant slip that remains in the critical cross section after unloading and also increases the initial drift rendering the post repair response more compliant; thus the chord drift at apparent yielding θy

R is greater than the initial θy

init. Note that the drift at apparent yielding θyR is attained when strain reaches the apparent yield

threshold at the post repair loading phase, εsyR=εs,pl

init (Figs. 1b, 2). The residual inelastic strain εsres is

represented by the dashed area depicted in Fig. 2 and it is the result of the difference between the strain distribution at apparent yielding (post repair) and the strain distribution at yielding (prior-to-repair loading when the element is in pristine condition). This paper aims to demonstrate the effect of yield penetration on column deformation capacity and on the build-up of residual deformation under cyclic loading. The strain development capacity of the anchorage is evaluated by deriving closed form solutions for the state of strain and slip along an anchorage, accounting for several design and behavioural parameters known to influence this problem, such as bond plastification and debonding of the cover after bar yielding, fracture energy release, and kinematic interactions between the bar ribs and the cover. From the derived expressions, the necessary conditions for FRP-jacketing of plastic hinge regions to be a sustainable solution in post-earthquake field applications are established. It is shown from first principles that the accumulated damage in the anchorage and lap splice zones of such structural elements reduces their available post-repair ductility and inelastic deformation capacity even after jacketing. The proposed expressions are also compared with the empirical expression for strain-capacity of anchored reinforcement (by EC8-III (2005) and the Greek Design Code for seismic rehabilitation and retrofit of concrete structures (KANEPE 2012).

Figure 1. a) Comparative representation between the initial and the repaired (with FRP) response of a lightlyreinforced column (definition of drift indices). b) Stress–strain law of tension steel (definition of strain indices).

Δεs,pl

rupture

εsy

εsr

fsy

non-usable strain of the bar due to limit in the available strain development capacity

fsu

εsoinit=εsy

R εso

max

εs,plinit

FRP- repair

b)

Pu

θyinit θu

init

80%Pu

θumax θy

R

Initial θpl

init

θplR Δθpl

R

a)

εsres

Es

Esh

θplmax

ℓrinit ℓp

init

fbmax

εεeell εεssyy

fbres

εsyR=εso

init

εεeell εεssyy fb

max

ℓpy

Repaired condition: apparent yielding

Pristine condition: initial yielding

Lb

εsyR=εso

init

εεssyy

Dashed area: strainresponsible for residualslip after unloading

Figure 2. Distribution of strain andbond along the anchorage length uponinitial yielding and apparent yielding.

2. KINEMATIC RELATIONSHIPS DURING INELASTIC RESPONSE 2.1 Basic bond assumptions for straight anchorages The approach adopted for the derivation of the kinematic relationships presented below is based on the mechanics of bond and development capacity of reinforcement. In order to evaluate any strain level above the yield point of an undisturbed straight anchorage or even the strain development capacity of a previously damaged anchorage, the solution of the basic field equations of bond is required; these equations establish force-equilibrium of an elementary bar segment and enforce compatibility between bar slip and axial bar strain. Necessary assumptions are an elasto - plastic (with hardening) stress-strain relation for steel reinforcement (Fig. 1b) and a simplified elastic-plastic bond-slip law followed by a post-peak residual plateau (Fig. 3). The solution for the straight anchorage below, is also applicable to an anchorage with a hook, which mathematically is treated as a straight anchorage of an equivalent length Lb,equiv (Fig. 3c). The equivalent length may be calculated considering the force capacity of an embedded hook. According to the Model Code 2010 this force is Fh=50fbAb where fb the bond strength and Ab the cross section area of the bar; thus the total bond force is Ftot=Fstraight+Fh or πDbfbLb,equiv=πDbfbLb,straight+50fbπDb

2/4 which results in Lb,equiv=Lb,straight+12.5Db. The term 12.5Db is multiplied by the ratio of design bond strength of ribbed to smooth bars (fb,rib/fb,sm) in order to be also valid for smooth reinforcing bars. 2.1.1 Prior repair loading phase For the initial loading phase and for inelastic strain εso

init greater than the yield strain εsy at the loaded end of an intact anchorage the solution of the field equations results in the following expressions that define the strain, slip and bond values at the knee points of the corresponding distributions along the embedded length (Tastani and Pantazopoulou, 2012). Bond assumes its residual resistance fb

res, strength fb

max and attenuation along the anchorage as per the distributions shown in Fig. 3a; superscript “init” refers to terms associated with the initial phase of loading.

- The strain εsoinit and slip so at the critical cross section (loaded end of anchorage where fb=fb

res) are:

( ) )4/(4 resbshbsy

initso

initr

initr

shb

resb

syinitso fED

EDf

⋅−=⇒⋅+= εεεε ll (2.1)

Lb

Figure 3. Based on the assumed bond law, attenuations of bar strain, slip and bond along the anchorage arederived after bar yielding at a) the end of the initial loading phase and b) at the maximum anchorage straincapacity after repair with FRP jacketing. c) Anchorage with a hook is treated as straight of equivalent length.

θ slip s1 s2

s

fb fbmax

fbres

Ls: shear span

ℓrinit ℓp

init

so s1 sr

sf

fbmax

εεeell εεssyy

fbres

εsoinit

> εsy

ℓrr=Lb- Lb,minLb,min

εsomax

fb

max

ℓrr

somax

sr=s2 s1

fbres

εεssyy

s3 Initi

al lo

adin

g A

fter-

repa

ir lo

adin

g

a)

b)

c) Lb,straight  

Lb,equiv

( ) initrsy

initsoro ss l⋅++= εε5.0 (2.2)

where ℓrinit

=yield penetration length, Esh =hardening modulus of steel, sr =slip at the end of the yielded segment, ℓr

init (measurements start from the loaded end, Fig. 2), Db=bar diameter.

- The strain εel at the end of ℓpinit (this is the length of anchorage adjacent to ℓr

init, over which bond has attained the plasticity limit fb

max), and slip sr at the end of ℓrinit and thereby at the start of ℓp

init are:

initp

bs

bsyel DE

fl⋅−=

max4εε (2.3)

( ) initpelsyr ss l⋅++= εε5.01 (2.4)

Parameter Es is the elastic modulus of steel. The strain εel that corresponds to the initiation of the elastic response of bond along the remaining length, Lb-(ℓr

init+ℓpinit), is defined through the elastic

solution (Tastani and Pantazopoulou, 2012) and is given by Eq. (2.5):

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⋅=

−−−−−− )(2)(21 1/1

initp

initrbLinit

pinitrbL

el ees llll ωωωε (2.5)

Where, ω=[4fbmax/(DbEss1)]0.5 is a measure of bond stiffness (fb

max/s1), as compared with that of steel, Es. Given the strain at the anchorage entrance, εso

init, the system of Eqs. (2.1 – 2.5) is solved for the definition of the unknown parameters (so, sr, εel , ℓr

init and ℓpinit). Note that by combining Eqs. (2.3-2.4)

a quadratic equation for ℓpinit is obtained as follows: 2fb

max/(EsDb)·(ℓpinit)2-εsy· ℓp

init+(sr-s1)=0, acceptable root of which is the one that also satisfies the requirement of ℓp

init+ℓrinit≤Lb and is given by Eq. (2.6):

( ) ( )( )sbbsbrbsysyinitp EDfEDssf /4/)(8 max

1max2

⎟⎠⎞⎜

⎝⎛ −−−= εεl (2.6)

The slip at the loaded end of the anchorage, so, is rewritten by combining Eq. (2.2) and the quadratic equation above; for simplification of several terms, the definition of Lb,min=(Db/4)·(Esεsy /fb

max) is used.

( ) initso

initrsyb

initp

initr

initpo Lss εε ⋅+⋅−++= llll 5.02)(5.0 min,

21 (2.7)

For the calculations to be easier the minimum anchorage length Lb,min is expressed as Ψ·Db where parameter Ψ accounts for different strength categories of steel and concrete as shown in Table 1.

An important issue of the algorithm is the definition of local bond – slip law (Fig. 3), and specifically, the coordinates of the milestone points of the curve in Fig. 3. Based on research done by the authors (Tastani and Pantazopoulou 2010) bond enters the state of plastification after bar translation by s1=0.2-0.3mm for bars with a normal rib area in the order of fR=0.07. Slip s2 at the end of the horizontal branch of the bond law is not an intrinsic property of the bar-concrete interface as it mainly depends on the anchorage length. It takes on its maximum value for anchorage lengths greater that the minimum length Lb,min and when the strain attains the capacity εso

max. Bond strength fbmax of anchorages

confined with stirrups and /or FRP jackets (Tastani and Pantazopoulou 2007) is defined as:

)( ,,'max 233.02

bb

effffj

bb

yststt

bb ND

EtSND

fAf

DCf

εζ

πμ

++⋅= (2.8)

where μ is the coefficient of friction along the splitting plane (0.9-1.2 for ribbed, and 0.3 for smooth reinforcement), ζ accounts for the tensile behavior of the concrete cover (1 for fully elastic and 2 for

Table 1. Definition of Ψ for several steel & concrete qualities, units: MPa (fib Model Code2010: the local bond strength is fb

max=1.25∼2.5√fc for pullout failure and for strain ≤εsy; the lower end refers to good and the upper end

to improved bond conditions). For anchorages with hooks/bends Ψ values correspond to the straight segment. fc

’ 12 16 20 25 30 fb

max 8.7∼4.3 10∼5 11.2∼5.6 12.5∼6.3 13.7∼6.8

f sy 220 7∼13 6∼11 5∼10 5∼9 4∼8

400 12∼23 10∼20 9∼18 8∼16 8∼15 500 15∼29 13∼25 12∼23 10∼20 10∼19

fully plastic), C is the cover thickness, ft’ is concrete’s tensile strength (0.35~0.5√fc

’), Nb is the number of tension bars restrained by the stirrup legs included in Ast (Ast is the cross sectional area of stirrups crossing the splitting plane), S is the stirrup spacing, fst,y the stirrup yield stress, tj, Ef and εf,eff are the FRP jacket thickness the material modulus of elasticity and its effective strain as a mechanism of confinement for laps (in the range of 0.0015∼0.002). This term is omitted in the pre-repair loading phase. In anchorages that have been split lengthwise prior to FRP jacketing, the development capacity after the addition of the FRP is only marginally improved unless cover replacement has preceded the FRP jacketing. Thus, prior cover longitudinal cracking eliminates the first term in Eq. (2.8), with commensurate implications on the strain development capacity of the anchorage. The residual bond strength fb

res that is developed over the yield length of the anchorage may be calculated using Eq. (2.8) by assuming that the frictional coefficient μ reduces to a residual value in the order of μres=0.4~0.6 as long as the ultimate slip so of the critical section does not exceed the rib spacing s3; beyond this threshold μres is taken equal to zero. (Actually, over the part of the anchorage where yield penetration occurs, reinforcement to concrete bond is negligible due to the zero stiffness of steel, except for the clamping force exerted by transverse reinforcement at the discrete locations of contact between stirrups and the main bars, Timosidis and Pantazopoulou 2009.) 2.1.2 Post repair loading phase When the bar axial strain attains its peak value εso

max at the critical section of the column, the associated flexural moment is maximized thereby imposing a commensurate demand in terms of stress and strain on the tension reinforcement. The bar strain attenuates to zero at the bar free end (in the end of the anchorage) whereas slip at that point approximates the characteristic value s1 of the bond – slip law (Fig. 3b). At imminent failure of the bar by pullout, it is assumed that yielding has penetrated deep enough into the anchorage so that the remaining bonded length Lb,min barely suffices to support the bar force (Fig. 3b). In that extreme situation of maximum attainable yield penetration, the residual bonded length must mobilize the bond strength of the material (fb=fb

max), so that: Lb,min=Db/4(fsy/fbmax ); thus

bond stress over Lb,min is constant and equal to the strength. It follows from equilibrium that bar stresses attenuate linearly over that segment (since fb(x)=dfs(x)/dx). Since the bar is elastic over this bonded part, bar strains are also linearly varying over the bonded length, whereas strains over the yielded portion, ℓr

R (where used, the superscript “R” denotes the post-repair loading phase) are linearly distributed because bond stress is constant and equal to the residual resistance fb

res, ranging from the value of εsy at the end of the Lb,min, to the value of εso

max at the critical section (Fig. 3b). Clearly, higher strains than εso

max (the value is limited by the maximum attainable length of yield penetration) cannot be sustained in the critical section, as incipient bond failure of the bar along its anchorage will occur. For lower strain values at the critical section, (εsy<εso<εso

max) the corresponding length of yield penetration ℓr

R will be estimated from the set of Eqs. (2.1) to (2.7). The strain capacity of the bar εso

max and the corresponding slip designated at the anchorage entrance so

max are given from Eqs. (2.9) to (2.10) and derived from the Eqs. (2.1) to (2.7) by substituting where ℓp

R=Lb,min. The region of maximum sustainable yield penetration is denoted as ℓrR=Lb-Lb,min (Fig. 3b), a

part of which is the initial penetration length ℓrinit. Note that Eq. (2.5) results to zero at both sides when

substituting ℓrR=Lb-Lb,min and ℓp

R=Lb,min (use of superscript “R” instead of “init”).

( ) ( )shbres

bbbsyso EDfLL /4 min,max −⋅+= εε (2.9)

{( ) ( )( ) ( )

444444444 3444444444 21443421ceresisbondresidual

bbshbres

bbbsy

tionplastificabond

sybtail

o LLEDfLLLsstan

min,min,min,1max /25.0 −⋅−⋅+++= εε (2.10)

Combining Eqs. (2.9 - 2.10) the ultimate slip may be rewritten as:

( ) maxmin,1

max 5.05.0 sobbsybo LLLss εε ⋅−++= (2.11)

As previously noted, the slip at the end of the horizontal branch of the local bond-slip law s2 assumes its maximum value for anchorage lengths greater than Lb,min and when the bar attains its dependable strain capacity εso

max; thus, s2 comprises the first two parts of Eq. (2.10) as: s2 =s1+0.5Lb,minεsy. This is also highlighted in Fig. 3b through the strain distribution plotted below the anchorage: the slip s2 is the sum of the tail value s1 and the result from integration of strain εsy over the ℓp

R=Lb,min. Exploring the

variables included in Eq. (2.9) it is concluded that the strain capacity of the anchorage apparently depends on how deeply the yield penetration has proceeded as expressed by the term (Lb-Lb,min) as well as by the residual bond resistance fb

res and is inversely affected by the steel hardening modulus; stiff stress-strain response of the reinforcement after yielding reduces the strain capacity of its anchorage for a given development length Lb. Regarding Eq. (2.10) the ultimate slip at bar entrance so

max is the result of three contributions; (i) the slip of the anchorage tail – at imminent anchorage failure, this is taken equal to the slip at the end of the elastic part of the bond law, (ii) the slip due to integration of the linearly varying bar strains - from zero at the anchorage tail, to the yield strain εsy along the minimum length Lb,min and (iii) the slip due to yield penetration over the debonded length (Lb-Lb,min). Based on Eq. (2.9), the estimated strain ductility capacity, μεs=εs/εsy of steel reinforcement in tension, is plotted in Fig. 4, against the required anchorage length for two material categories ( S400 and S500) and five concrete qualities (C12, C16, C20, C25 and C30). (The curves are plotted with diminishing line thickness, where, the lower the concrete quality the thinner the corresponding curve, as shown in Figs. 4a,b). Note that in a complete structural member, the expected bar inelastic strain for a selected anchorage length may also be limited by other failure modes that precede anchorage failure. Here, residual bond strength fb

res (attained after bar yielding) was estimated according with the local bond model for εso>εsy, as prescribed by the fib Model Code 2010. Thus, fb

res=fbm=fbmax·Ωy where parameter

Ωy accounts for the bond deterioration as a function of the attained inelastic strain (Ωy=1-[0.85(1-e-

5ab)], a=(εs-εsy)/(εsr-εsy), b=[2-ft/fsy]2, where ft the bar stress at the assumed inelastic strain). Apart from

the steel and concrete qualities, other variables studied were, the hardening modulus of steel (1% and 5% of elasticity modulus Es) and the bond conditions (fb

max was taken equal to 2.5√fc’ and 1.25√fc

’ for good and all other bond conditions respectively, values taken from Table 1). Diagrams such as those

S400, Esh=1%Es

S400, Esh=5%Es

C30…C20…..C12Good bond cond.

All other bond cond.

C30…C12 C30…... C16 C12

All other bond cond. Good bond cond.

Figure 4. a) Bar axial strain ductility development capacity versus required anchorage length for steel S400,Esh=1%Es and 5%Es, concrete qualities: C12, 16, 20, 25, 30, and two bond conditions.

S500, Esh=1%Es

Figure 4. b) Bar axial strain ductility capacity, μεs against required anchorage length Lb/Db for steel S500,Esh=1%Es, concrete qualities: C12, 16, 20, 25, 30, and two bond conditions. c) Ductility μεs of a straight lapsplice: correlation with the EC8 (2005).

C30…... C16 C12

All other bond cond.

C30…C12

Good bond cond.

S400, Esh=1%Es

εsr=0.08 C25

f b=f b,

equi

v.=

2.5

ℓ ou,

min

(f b=

1.3M

Pa)

f bres =

1.3M

Pa

b) c)

plotted in Figs. 4a,b can be used to assess the reinforcement strain development capacity and thus, the rotation capacity at the critical sections of frame members. Note that according to Eurocode 8 (EC 8, 2005) and in the case of straight lapped bars starting at the critical cross section, the bar is able to develop its full strain capacity εsr if the available lap length ℓo is greater than a lower bound ℓou,min=Dbfsy/[(1.05+14.5αlρsxfst,y/fc

’)√fc’], else the lap deformability is

limited by multiplying the full strain capacity with the ratio, ℓo/ℓou,min. Parameter αl is the confinement effectiveness of prismatic members, equal to (1-sh/(2bo))(1-sh/(2ho))nsertr/ntot where nrestr= number of lapped bars laterally restrained by stirrup corner or cross-tie and ntot= total number of lapped bars along the cross section perimeter and ρsx=Ast/(hS) the ratio of transverse steel parallel to the direction of loading. When considering the design expression for the anchorage length, Lb= Dbfsy/(4fb), it follows that the required average bond strength for the bar to develop its full strain capacity would be fb=0.25·[(1.05+14.5αlρsxfst,y/fc

’)√fc’]. For a non-confined (no stirrups) lap-spliced bar with fsy=400MPa

(Es=200GPa and Esh=2GPa) and concrete strength fc’=25MPa the required average bond strength

would have to be fb=1.3MPa in order to attain the strain capacity of εsr=0.08 over a length ℓou,min of at least 76Db (dashed blue line in Fig. 4c). Alternatively, using the proposed algorithm for μεs=εs/εsy =0.08/0.002=40, it follows that Lb/Db=56 (dashed green line in Fig. 4c); note that in this case, at the onset of failure, along the last part of ℓp/Db=Lb,min/Db=16, bond would be fb

max=6.25MPa (for concrete C25, black curve in Fig. 4c, see also Table 1) whereas along the first part, ℓr/Db=(Lb-Lb,min)/Db=40 the residual bond strength would be fb

res=1MPa; the equivalent average bond strength is fb,equiv.Lb/Db=(fb

maxℓp+ℓrfbres)/Db which results to fb,equiv =2.5MPa > fb=1.3MPa.

2.2 Definition of inelastic drift components As was previously stated in the Introduction, the initially developed inelastic strain εso

init corresponds to the apparent yielding strain εsy

R in the post repair loading phase and thus the associated slip so (from Eq. (2.2)) increases the nominal drift at yielding from its initial value, θy

init, (solved from Eq. (2.12a), using Eqs. (2.1) to (2.7) and substituting where εso

init=εsy, ℓrinit=0 and ℓp

init=ℓpyield) to an apparent value

θyR in the post-repair loading phase. To account for this, the following Eq. (2.12b) is proposed. Note

that Eq. (2.12b) is valid when no measures are taken to mitigate bond damage before FRP jacketing.

ybyieldp

yieldps

inity

slip

sybyieldp

yieldp

flexural

ysyslipy

flexy

inity

LH

jdLsH

ϕθ

εϕθθθθ

⋅⎟⎠⎞⎜

⎝⎛ −+=

⇒⎥⎦⎤

⎢⎣⎡

⎟⎠⎞⎜

⎝⎛ −++=⇒+=

min,2

min,2

1

26/

/26

ll

4444444 84444444 76

ll48476

(2.12a)

444444444444 8444444444444 76

llll48476

slip

initso

initrsyb

initp

initr

initp

flexural

ysslipy

flexy

Ry jdLsH /5.025.06/ min,

21 ⎟

⎠⎞

⎜⎝⎛ +⋅⎟

⎠⎞⎜

⎝⎛ −+++⋅=+= εεϕθθθ

initu

initryb

initp

initr

initps

Ry LH ϕϕθ llll 5.025.06/ min,

2+⋅⎟

⎠⎞⎜

⎝⎛ −++=⇒ (2.12b)

Term s1/jd above in Eqs. (2.12) is a very small fraction of the total drift and so it has been omitted. Variable Hs is the column free height (it has been assumed here that during lateral sway the column bends in double curvature), jd is the distance of the tension steel to the centroid of the compression

zone c (considering, as a simplifying approximation, that the depth of compression zone does not change significantly after yielding: it is usually taken as 0.9d, where d the cross section’s effective depth), φy is the curvature of the critical cross section at yielding and φu

init is the inelastic curvature developed in the initial loading phase. For the definition of φy=εsy/(d-c) the parametric diagram of Fig. 5 (Thermou et al. 2012) is used: given the total reinforcement ratio of a cross section and the imposed axial load, the normalized compression zone depth at section yielding, c/d, is defined. For lightly reinforced columns (ρtot≤1%) and axial load less than

Tot

al r

einf

. are

a, ρ

tot

c/d

Figure 5. Compression zone depth as afunction of reinforcement ratio and axial load.

0.4fc’Ag the c/d ranges between 0.23-0.28.

The strain demand εso>εsy developing in the critical section of a column experiencing relative lateral drift as part of the lateral load resisting system of any structure is interpreted through the kinematics of the deformed member. (The relationships below are applicable both for columns tested in pristine condition or after repair; thus, for simplicity, they are presented without superscripts “init” or “R”.) Thus, if θu the total chord rotation of the member, then it may be easily shown from concrete mechanics that the inelastic curvature φu (or the plastic curvature, φu-φy) at the critical section is related with θu =θu

flex+θuslip as follows (by also using Eq. (2.7)):

( )yur

yb

ppru

ry

b

prp

slipu LL

ϕϕϕϕϕθ −+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+=+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+=

22)(

22)(

5.0min,

2

min,

2ll

llll

ll

( ) ( ) ⇒−+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−++−+=+=

4444444 84444444 76

llll

4444 84444 76

44 344 21l

43421

slip

yur

yb

ppr

flexural

plastic

yuspanp

elastic

ysslipu

flexuu L

H ϕϕϕϕϕϕθθθ

θθ22

)(6

min,

2

( )yurspan

pyb

prp

su L

Hϕϕϕθ −⎟

⎠⎞

⎜⎝⎛ ++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−++=

22)(

6 min,

2l

ll

ll (2.13)

The ultimate drift that can be supported by the anchorage may be calculated by taking into account the ultimate slip from Eq. (2.11) as follows (note that φu

max=εsomax/jd, εso

max is taken from Eq. (2.9)):

( ) ( )444444 8444444 7644444 844444 76

l

slip

ubbyb

flexural

yuspanpysu LLLH max

min,maxmax 5.05.06/ ϕϕϕϕϕθ −++−+⋅= (2.14)

In Eqs. (2.13-14) plastic hinge length ℓpspan is assumed 0.5h for ribbed bars and 0.25h for smooth bars.

Material strains at the critical cross section may be estimated from a first-order approximation; this along with the plane sections assumption enables estimation of strains on any point when either of the two materials has reached a milestone in its stress-strain relationship; thus, strain in the tension steel at the critical cross section, εso, and in the extreme compression fiber of concrete, εc,c, are related as per:

( ) ccuso ccdcd ,)( εϕε ⋅−=⋅−= (2.15)

Considering, as a simplifying approximation that the depth of compression zone does not change significantly after yielding, it follows that (after substitution of (ϕu-ϕy) from Eq. (2.13)):

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−++−

+−

=−⋅−=−= yb

prp

su

rspanp

yusysopls LHcdcd ϕθϕϕεεε

min,

2

, 2)(

65.0)()()(

lll

ll (2.16)

Equation (2.16) states that the plastic strain, εs,pl (Fig. 1b), which occurs in the tension reinforcement is

linearly related to the inelastic drift θu experienced by the member. Another conclusion from the above analysis is that the larger the length of yield penetration ℓr, the larger the fraction of the total rotation θu that is owing to elastic strains φy, and the smaller the fraction that remains for plastic rotation. The latter is also downsized by the length of the plastic hinge due to inelastic strain in the shear span. So although the total rotation capacity may be high (i.e. in the order of 4-6%), in the presence of significant yield penetration, response may be unacceptable owing to the very high compliance (flexibility) of the member to lateral displacement without toughness. Furthermore, there is a limiting εs,pl

max value that may be sustained by the anchorage before pullout failure (that should also be limited by the plastic rupture threshold of steel, εsr-εsy, where εsr≈0.08 for Db>6mm). Thus from Eq. (2.9) it is:

( ) sysrbbshbres

bsyopls LLEDf εεεεε −≤−⋅=−= min,maxmax

, )/(4 (2.17)

Equations (2.1) to (2.17) are also valid with regards to lap-splices developed over ℓo within the plastic hinge region (where Lb=ℓo and Lb,min=ℓo,min), whereas the plastic hinge length in the span ℓp

span is equal to zero. Note that after the preloading phase (i.e., when the initial damage is inflicted), if the imposed

drift has exceeded even once the yielding limit, then upon unloading the tension reinforcement has residual plastic strains at the critical section (Fig. 1b), their magnitude given by Eq. (2.16). But of far greater significance is that the residual strain capacity of the bar, which is available for inelastic response during the post-repair service-life of the member, is the difference of Δεs,pl=εs,pl

max-εs,plinit (Fig.

1b). Therefore, in the new phase of loading (after retrofit), the peak inelastic drift that may be imposed on the specimen before deterioration due to anchorage failure is only:

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−++−

+−

−−=Δ yb

initpinit

rinitp

suinit

rspanpshb

resb

bbpls LHcd

EDfLL ϕθε

min,

2

min,, 2)(

65.0)(4 l

llll

(2.18)

Equation (2.18) identifies the fundamental difference in FRP-jacketing of pre-damaged columns as compared to columns in pristine condition – for columns that have experienced excessive yielding prior to jacketing, the plastic rotation capacity after jacketing is limited by the dependable residual strain range of tension reinforcement, which depends on the extent of damage that has been sustained in the anchorage – i.e. over a part of the structure that lies outside the jacketed region. 3. ALGORITHM VERIFICATION: AN EXPERIMENTAL CASE STUDIE  To demonstrate the implications of previous damage on the performance of FRP-jacketed columns and the applicability of the proposed algorithm, two relevant experiments are used for this purpose. These are the as built specimen Ls2_b by Syntzirma et al. 2006, which was subsequently repaired with CFRP jacketing, denoted by the code name RcLs2_b by Thermou and Pantazopoulou ’09, Table 2. The specimens were cantilevers having a clear height of 900mm, a 200mm square cross section, reinforced with 8 spliced pairs of bars placed uniformly on the perimeter of the cross section and clear cover of 20mm; rectangular stirrups were spaced at 70mm o.c. One bar of each pair was anchored in the footing having a straight length of 300mm and a U-shaped hook of total length 900mm; however the equivalent length was set to Lb,equiv=300+12.5Db=450mm. The column had been originally loaded under a reversed cyclic displacement history with a combined constant axial load of 0.08fc

’Ag up to splitting failure along the splice length. After the initial test, the specimen was repaired by cover replacement with grout and CFRP jacketing and was subsequently reloaded up to failure by excessive pullout of reinforcement localized at the connection between the column and the footing.

Initial loading phase (Ls2-b): fbmax=6.72MPa (Eq. (2.8)), fb

res=3.36MPa, ℓo,min=257mm, effective depth d=174mm and depth of compression zone c=50.5mm (from Fig. 5). Based on Eq. (2.12-a) the initial chord drift at yield (ℓp

yield=149mm, ϕy=0.0000233mm-1, εsy=576MPa/200GPa=0.00288) is θyinit=1.07%,

a magnitude very close to the experimental measurement θy,exp=1.10%. Based on the drift value attained at termination of the initial experiment by the as-built specimen (θu,exp=1.76%) and using back-calculation (Eq. (2.13): θu

init=θu,exp=1.76%) the corresponding curvature is resolved: ϕu

init=0.00021mm-1 (ℓrinit=26mm, ℓp

init=170mm) and the consumed plastic strain (Eq. (2.16)) is εs,pl

init=0.0236, i.e., less than the εs,plmax=0.039 (from Eq. 2.17) <εsr-εsy =0.08-0.00288=0.077. Thus, if

no other mode of failure prevails, the splice will have a dependable strain capacity in the post-repair phase, equal to Δεs,pl =εs,pl

max -εs,plinit =0.039-0.0236=0.0154. In case where no splice repair occurs the

θu,exp becomes the apparent chord drift at yielding upon repair, θyR (as per Eq. (2.12b)); thus, it would

be expected that θyR =1.76%. This magnitude is very close to the experimentally measured drift at

yielding of the repaired specimen θy,exp=1.74% (Table 2) despite the fact that the specimen had been repaired in the splice region (new cover and CFRP jacket) and thus the calculated value for θy

R should be equal to θy

init. This inconsistency implies that a significant share of slip comes from the anchorage in the footing, which did not undergo any form of repair intervention in the second loading phase. Loading phase after repair (RcLs2-b): After retrofitting of the splice region, the bond strength over the jacketed length is increased to fb

max=12.7MPa (fbres=6.4MPa) owing to confinement provided by the

Table 2. Deformation and strength indices of the studied specimens

Name lrs ρfv (%)

θy exp (%)

θu,exp (%)

θ80%u (%)

Py (kN)

Pu (kN)

Ls2-b No --- 1,10 1,76 3,50 27 34 RcLs2-b 5-C 1,10 1,74 4,72 6,93 45 45

steel: longitudinal Db=12mm, fsy=576MPa, fr=670MPa, stirrups: Dst=6mm, fst,y=335MPa, fst,r=432MPa, concrete fc

’=20MPa, CFRP: tf=0.11mm, Ef=230GPa, εf,u=0.015, ℓo=300mm (ℓo/Db=25).

CFRP jacketing (5 plies) and the contribution of the new cover with result being the reduction of ℓo,min=137mm (as compared with the initial value of ℓo,min=257mm). If the specimen was strengthened (without initial damage) rather than retrofitted then its yield drift should be only θy

init=0.91%. According to the analysis the applied repair scheme would be sufficient for the splice to develop the full strain capacity of the reinforcement (limited by the steel rupture strain, so εso

max=εsr=0.08.) Thus from back-calculation (Eq. (2.13)) it follows that θu

R=θu,exp=4.72% and the corresponding curvature, ϕu

R=0.00062mm-1, with ℓrR=43mm, ℓp

R=41mm and the consumed plastic strain (Eq. (2.16)) is εs,pl

R=0.074 which approximates the rupture limit εsr-εsy=0.077. According to the experiment the loading of the specimen was stopped after displacement ductility μΔ=5 (most of the inelastic rotation at that point was owing to slip concentrated in a single crack) whereas strength loss was in the range of 20% which is the point of theoretical failure. This experimental response is confirmed by the analysis where the bar at this level of drift had consumed its strain reserves (some specimens from the same experimental series exhibited bar fracture after load reversal). 4. CONCLUSIONS Retrofits of initially damaged brittle r.c. columns through FRP jacketing in the plastic hinge zones fall within two categories when considering the damage attained during the pre-repair phase; the first includes columns that had attained during first loading a premature mode of failure without ductility, whereas the second refers to columns that had developed inelastic strains in the critical section. In the latter case the contribution of FRP jacketing is less efficient in terms of strain development capacity of reinforcement due to yield penetration into the anchorage which destroys interfacial bond. The strain development capacity of the jacketed anchorage is evaluated by closed form solutions of the field equations of bond over the embedded length considering bond plastification, cover debonding after bar yielding and spread of inelasticity into the anchorage. From the derived expressions, the necessary conditions for FRP-jacketing of plastic hinge regions to be a sustainable solution in post-earthquake field applications are established. It is shown from first principles that the accumulated damage in the anchorage and lap splice zones of such structural elements reduces their available post-repair ductility and inelastic deformation capacity even after jacketing. The proposed expressions are also compared with the empirical expression for strain-capacity of anchored reinforcement that is recommended by EC8-III (2005) and the recent Greek Design Code for seismic rehabilitation (KANEPE 2012). 5. REFERENCES Eurocode 8 (EC8-III), ‘Design of structures for earthquake resistance - Part 3: Assessment and retrofitting of

buildings’, EN1998-3-2005:E 2005, European Committee for Standardization (CEN), Brussels. FIB Model Code 2010 for Concrete Structures (SAG 5). Draft edition. Greek Design Code for Seismic Rehabilitation and Retrofit of Concrete Structures (KANEPE 2012).

www.oasp.gr Syntzirma D., Thermou G., Pantazopoulou S., Halkitis G. (2006). Experimental research of r.c. elements with

substandard details. 1st European Conf. on Earthquake Eng. and Seismology, Geneva, Switzerland, 2006, (paper # 819).

Syntzirma, D.V., Pantazopoulou, S.J. (2006). Assessment of deformability of r.c. members with substandard details. 2nd International fib Congress, Naples, Italy, 2006 (paper number 446).

Tastani S.P., Pantazopoulou S. J. (2007). Detailing procedures for seismic rehabilitation of reinforced concrete members with fiber reinforced polymers. Elsevier Engineering Structures 30:2, 450-461.

Tastani S.P., Pantazopoulou S.J. (2010). Direct tension pullout bond test: experimental results. ASCE J. of Structural Engineering 136:6, 731-743.

Tastani S.P., Pantazopoulou S.J. (2012). Modelling reinforcement to concrete bond. ASCE J. of Structural Engineering (under review).

Thermou G., Pantazopoulou S. (2009). Fiber reinforced polymer retrofitting of substandard r.c. prismatic members. ASCE J. Composites for Construction 13:6, 535-546.

Thermou G.E., Pantazopoulou S.J., and Elnashai A.S. (2012). Global Interventions for seismic upgrading of substandard RC buildings. ASCE J. of Structural Engineering 138:3, 387-401".

Timosidis D., Pantazopoulou S.J. (2009). Anchorage of longitudinal column reinforcement in bridge monolithic connections. ASCE Journal of Structural Engineering 135:4, 344–355.