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Flexural Behavior of Brittle RC Members Rehabilitated

with Concrete Jacketing

G. E. Thermou, Ph.D.1; S. J. Pantazopoulou, M.ASCE2; and A. S. Elnashai, F.ASCE3

Abstract: The composite ﬂexural action of prismatic reinforced concrete 共RC兲members repaired/strengthened by RC jacketing was

modeled with a dual-section approach. The model considers the relative slip at the interface between the existing member and the jacket

and establishes the mechanisms that are mobilized to resist this action, thereby supporting composite behavior. An iterative step-by-step

incremental algorithm was developed for calculating the overall ﬂexural response curve. Consideration of frictional interlock and dowel

action associated with sliding at the interfaces as well as the spacing and penetration of ﬂexure-shear cracks are key aspects of the

algorithm. The proposed procedure was veriﬁed through comparison with published experimental data on RC jacketed members. The

sensitivity of the upgraded member’s ﬂexural response to jacket design variables was investigated parametrically. Monolithic response

modiﬁcation factors related to strength and deformation indices were evaluated and the sensitivity of the model was assessed.

DOI: 10.1061/共ASCE兲0733-9445共2007兲133:10共1373兲

CE Database subject headings: Concrete, reinforced; Rehabilitation; Seismic design; Inelasticity; Flexure.

Introduction

Reinforced concrete jacketing is a traditional method for seismic

upgrading of damaged or poorly detailed reinforced concrete con-

struction. In applying this technique, the objective is to suppress

alternative premature modes of failure that would otherwise pre-

vail in the structural members under reversed cyclic loading,

thereby promoting ﬂexural yielding of primary reinforcement.

Through reinforced concrete 共RC兲jacketing stiffness and strength

are increased, whereas dependable deformation quantities may or

may not be enhanced, depending on the aspect ratio of the up-

graded element and the factors limiting deformation capacity in

the initial state of the element. For practical purposes, response

indices of the jacketed members such as resistance and deforma-

tion measures at yielding and ultimate are routinely obtained by

applying pertinent multipliers on the respective properties of

monolithic members with identical geometry. The multipliers are

referred to in the literature as monolithic factors,Ki.

Depending on the member property being scaled 共strength or

stiffness兲, the method of load application and the jacket function,

various values have been reported for Ki, ranging from 0.7 up to

1. Eurocode 8 共CEN 1996兲recommends KR=0.8 for strength and

KK=0.7 for stiffness provided that: 共1兲Loose concrete and buck-

led reinforcement in the damaged area have been repaired or re-

placed before jacketing; 共2兲all new reinforcement is anchored

into the beams and slabs; and 共3兲the additional concrete cross

section is not larger than twice the cross section of the existing

column. Based on the results of a recent experimental study con-

ducted by Vandoros and Dritsos 共2006a,b兲, the monolithic factors

associated with strength, stiffness, and deformation vary greatly

depending on the techniques followed in constructing the jacket.

For example, it was shown that dowels improve the ductility ca-

pacity of the jacketed member, roughening of the interface in-

creases the energy absorption capacity, and a combination of the

two procedures improves stiffness.

Monolithic factors are used by codes of practice for con-

venience, as the mechanics of composite action of jacketed re-

inforced concrete members under cyclic shear reversals is too

complicated for practical calculations. So far the focus has been

on stiffness and strength, whereas no speciﬁc reference has been

made for monolithic factors related to deformation indices. A de-

tailed method for calculating these factors would be required in

order to assess their parametric sensitivity to the relevant design

variables. From the available experimental evidence it appears

that slip and shear stress transfer at the interface between the

outside jacket layer and the original member that serves as the

core of the upgraded element are controlling factors 共CEN 1996;

KANEPE 2004兲. Indeed, sliding failure at the interface limits the

strength and affects the rotation capacity of the entire member.

This paper presents a detailed procedure for estimating the

behavior of concrete members jacketed with an outer RC shell.

The composite action that jacketed reinforced concrete members

develop in ﬂexure greatly depends on the force transfer that oc-

curs between the core and the jacket. Estimating strength and

deformation capacity of such members is a complex mechanics

problem that is hampered by the limited understanding of the

interfacial resistance mechanisms such as friction, interlock, and

dowel action. To calculate the monolithic factors and to establish

1Dept. of Civil Engineering, Laboratory of Reinforced Concrete,

Demokritus Univ. of Thrace, Vas. Soﬁas 12, Xanthi 67100, Greece.

E-mail: gthermou@civil.duth.gr

2Professor, Dept. of Civil Engineering, Laboratory of Reinforced

Concrete, Demokritus Univ. of Thrace, Vas. Soﬁas 12, Xanthi 67100,

Greece 共corresponding author兲. E-mail: pantaz@civil.duth.gr

3William J. and Elaine F. Hall Professor, Dept. of Civil and

Environmental Engineering, Univ. of Illinois at Urbana-Champaign,

2129 Newmark CE Lab., 205 North Mathews Ave., Urbana, IL 61801.

E-mail: aelnash@uiuc.edu

Note. Associate Editor: Dat Duthinh. Discussion open until March 1,

2008. Separate discussions must be submitted for individual papers. To

extend the closing date by one month, a written request must be ﬁled with

the ASCE Managing Editor. The manuscript for this paper was submitted

for review and possible publication on October 27, 2005; approved on

March 9, 2007. This paper is part of the Journal of Structural Engineer-

ing, Vol. 133, No. 10, October 1, 2007. ©ASCE, ISSN 0733-9445/2007/

10-1373–1384/$25.00.

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1373

their dependence on critical design variables an analytical model

is developed in this paper from ﬁrst principles. The signiﬁcance

of jacket detailing on the resulting response and the associated

values of the monolithic factors for strength and deformation ca-

pacity is demonstrated and quantiﬁed through parametric studies

and correlation of analytical estimates with test results.

Analytical Model for RC Jacketed Members

It is assumed that the existing member core is partially connected

with the external jacket layer, so that the mechanisms of force

transfer at the interface are mobilized by relative slip of the two

bodies. In analyzing the ﬂexural behavior, the cross section of the

upgraded member is divided into three layers. The two external

ones represent the contribution of the jacket, whereas the middle

layer represents both the core 共existing cross section兲and the web

of the jacket shell 关Fig. 1共a兲兴. For reference in the remainder of

this derivation, the Cartesian coordinate system is oriented so that

the xaxis is parallel to the longitudinal member axis, the yaxis is

along the cross-sectional depth, whereas the zaxis is oriented

along the cross-sectional breadth 关Fig. 1共d兲兴. The difference in

normal strain at the interface between layers accounts for the

corresponding slip in the longitudinal direction; thus, only the

implications of slip along horizontal planes are considered in the

model. The inaccuracy associated with neglecting shear transfer

along the vertical contact faces 共i.e., on faces normal to the zaxis兲

is small if jacket longitudinal tension reinforcement is evenly dis-

tributed in the perimeter 关Fig. 1共a兲兴. Note that in that case, a

vertical slice of the jacketed cross section is self-equilibrating

关consider for example the rectangular portion of the cross section,

to the left of line A-A’ in Fig. 1共a兲兴. This means that the total

stress resultant is zero since compression and tension forces over

the height of the segment are in equilibrium; hence the shear

stress xz 关Fig. 1共d兲兴 acting in a plane normal to the zaxis and

oriented in the longitudinal direction is also zero. As usually done

in ﬂexural analysis of layered composite beams, it is assumed that

the three layers deform by the same curvature, 关Fig. 1共a兲兴. From

free body equilibrium of any of the two exterior layers the shear

ﬂow at the interface is calculated as the difference in the stress

resultant between two adjacent cross sections. The procedure is

implemented in an iterative algorithm that employs dual-section

analysis. A key element of the algorithm is the shear stress slip

relationship used to describe the behavior of the interface between

layers.

Interface Shear Behavior

Slip at the interface between the existing member and the jacket is

explicitly modeled. Mechanisms that resist sliding are: 共1兲Aggre-

gate interlock between contact surfaces, including any initial ad-

hesion of the jacket concrete on the substrate; 共2兲friction owing

to clamping action normal to the interface; and 共3兲dowel action

of any pertinently anchored reinforcement crossing the sliding

plane. Thus, in stress terms, the shear resistance, vn, against slid-

ing at the contact surface, is

vn=va+vc+vD=va+N+vD共1兲

In Eq. 共1兲va⫽shear resistance of the aggregate interlock

mechanism; ⫽interface shear friction coefﬁcient; N⫽normal

clamping stress acting on the interface; and vD⫽shear stress re-

sisted by dowel action in cracked reinforced concrete. The ﬁrst

two terms collectively represent the contribution of concrete as

they depend on the frictional resistance of the interface planes.

The clamping stress represents any normal pressure, p, externally

applied on the interface, but also the clamping action of reinforce-

ment crossing the contact plane as illustrated in Fig. 1共b兲. From

equilibrium requirements it is shown

N=p+sfs共2兲

where p⫽normal pressure externally applied on the contact plane;

fs⫽axial stress of the bars crossing the interface; and

s⫽corresponding reinforcement area ratio.

Shear transfer is affected by the roughness of the sliding plane,

by the characteristics of the reinforcement, by the compliance

of concrete, and by the state of stress in the interface zone. Dowel

action develops by three alternative mechanisms, namely, by di-

rect shear and by kinking and ﬂexure of the bars crossing the

contact plane. A variety of models are available for modeling

the interface phenomena. In this study, the model developed by

Tassios and Vintzēleou 共1987兲and Vintzēleou and Tassios 共1986,

1987兲as modiﬁed by Vassilopoulou and Tassios 共2003兲was used

due to its simplicity and robustness. The model estimates the

combined dowel and shear friction resistances for a given slip

value at the interface, as follows:

1. Frictional resistance: The concrete contribution term in

Eq. 共1兲,vc共s兲, is described by the following set of equations:

vc共s兲

vc,u

= 1.14

冉

s

sc,u

冊

1/3

for s

sc,u

艋0.5 共3a兲

vc共s兲

vc,u

= 0.81 + 0.19

冉

s

sc,u

冊

for s

sc,u

⬎0.5 共3b兲

where sc,u⫽ultimate slip value beyond which the frictional

mechanisms break down 共sc,uis taken approximately equal

to 2 mm兲共CEB-FIP 1993兲. The normalizing term,

vc,u⫽ultimate frictional resistance of the interface, given by

vc,u=共fc

⬘2N兲1/3 共4兲

where ⫽ultimate interface shear friction coefﬁcient taken

equal to 0.4 and fc

⬘⫽concrete cylinder uniaxial compressive

strength 关Fig. 1共b兲兴. To calculate the axial stress of the bars

crossing the interface, fs, the separation wbetween contact

surfaces as they slide overriding one another is considered

关Fig. 1共c兲兴. According to Tassios and Vintzēleou 共1987兲the

Fig. 1. 共a兲Strain proﬁles; 共b兲normal stresses at interface; 共c兲pull-out

displacement of bars crossing interface; and 共d兲state of stress acting

on inﬁnitesimal element in initial coordinate system

1374 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007

separation wand lateral slip, s, are related by: w=0.6·s2/3.To

account for w, it is assumed that the bars pull out by w/2

from each side of the contact surface. Considering uniform

bond stresses along the embedment length, the axial bar

stress, fs, at the contact plane is estimated from

fs=

冉

0.3s2/3Esfc

⬘

Db

冊

1/2

共5兲

In Eq. 共5兲,Es⫽elastic modulus of steel; and Db⫽diameter

of the bars clamping the interface 共here, the stirrup legs of

the jacket兲.

2. Dowel resistance: In the dowel model the bar behaves as a

horizontally loaded free-headed pile embedded in cohesive

soil. Yielding of the dowel and crushing of concrete are

assumed to occur simultaneously. Dowel force 关the resultant

of term vDin Eq. 共1兲兴 is obtained from the relative interface

slip sas follows 关Fig. 1共c兲兴

VD共s兲

VD,u

= 0.5 2

sd,el

for s艋sd,el = 0.006Db共6a兲

for

VD共s兲

VD,u

艌0.5 ⇒s= 0.006Db

+ 1.76sd,u

冋

冉

VD共s兲

VD,u

冊

4

− 0.5

冉

VD共s兲

VD,u

冊

3

册

共6b兲

where sd,el⫽elastic slip value; sd,u⫽ultimate slip value;

VD,u⫽ultimate dowel force; and Db⫽diameter of the bars

offering dowel resistance 共here, legs of the jacket transverse

reinforcement兲.

In Eq. 共6b兲the dowel force, VD共s兲, is estimated iteratively

given the slip magnitude, s. The ultimate dowel strength and as-

sociated interface slip are given by

VD,u= 1.3Db

2共fc

⬘fsy共1−␣兲2兲1/2;sd,u= 0.05Db共7兲

where ␣⫽bar axial stress normalized with respect to its yield

value and fsy⫽yield strength of steel.

The total shear resistance of an interface with contact area Aint

crossed by kdowels is

V共s兲=vc共s兲Aint +kVD共s兲共8兲

where vc共s兲and VD共s兲are calculated from Eqs. 共3兲and 共6兲, re-

spectively, for a given amount of interface slip.

Estimation of Crack Spacing

Similar to conventional bond analysis, shear transfer at the inter-

face between the existing member and the jacket is carried out

between half crack intervals along the length of the jacketed

member 关Fig. 2共a兲兴. To evaluate the crack spacing the stress state

at the crack is compared with that at the midspan between adja-

cent cracks 关Fig. 2共b兲兴. It is assumed that at the initial stages of

loading cracks form only at the external layers 共jacket兲increasing

in number with increasing load, up to crack stabilization. This

occurs when the jacket steel stress at the crack, fs,cr exceeds the

limit 共CEB-FIP 1993兲

fs,cr ⬎fctm

1+s,eff

s,eff

共9兲

where fctm⫽tensile strength of concrete; 共=Es/Ecm兲⫽modular

ratio; and s,eff⫽effective reinforcement ratio deﬁned as the total

steel area divided by the area of mobilized concrete in tension,

usually taken as a circular domain with a radius of 2.5Dbaround

the bar 共CEB-FIP 1993兲. Using the same considerations in the

combined section it may be shown that a number of the external

cracks penetrate the second layer 共core兲of the jacketed member

关Fig. 2共a兲兴. From the free body diagram shown in Fig. 2共b兲the

shear ﬂow, qs, at the contact between the bottom layer and the

core is estimated as

qs=NjDb,J

bJ

fb,J共10兲

where NJ⫽number of bars in the tension steel layer of the jacket;

Db,J⫽bar diameter of the jacket longitudinal reinforcement;

fb,J⫽average bond stress of the jacket reinforcement layer; and

bJ⫽width of the jacketed cross section. The crack spacing is es-

timated from free body equilibrium in the tension zone of the core

of the composite section 关Fig. 2共b兲兴. Assuming that the neutral

axis depth is about constant in adjacent cross sections after stabi-

lization of cracking, the crack spacing is deﬁned as follows:

c=2bJlcfctm,c

NcDb,cfb,c+qsbJ

共11兲

where Nc⫽number of bars in the tension steel layer of the

core; Db,c⫽bar diameter of the core longitudinal reinforcement;

fb,c= average bond stress of the core reinforcement layer;

lc⫽height of the tension zone in the core component of the com-

posite cross section; and fctm,c⫽tensile strength of concrete core.

Fig. 2. 共a兲,共b兲Deﬁnition of crack spacing; 共c兲estimation of vertical shear stress xy, denoted here as vdi; and 共d兲rotation of jacketed cross section

due to slip

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1375

Shear Stress Distribution on Cross Section

of Jacketed Member

To analyze jacketed members in ﬂexure the composite cross

section is assumed to deform in its plane of symmetry with a

curvature ; relative slip occurs in the horizontal contact planes

between the top and bottom jacket layers and the core 关Fig. 1共a兲兴.

Section equilibrium is established and the normal stress resultant

of each layer, ⌺Fi, is estimated 关Fig. 2共c兲兴. Using dual-section

analysis 共Vecchio and Collins 1988兲, and considering that the

shear force at any section equals the moment gradient along the

member length, the layer stress resultant ⌺Fi, is used to calculate

the vertical shear stress demand of the member 关i.e., stress xy,

oriented in the yaxis in Fig. 1共d兲兴, at layer ith, denoted here by

the term vd,i, from

vd,i=⌺Fi

0.5cbJ

共12兲

where cis obtained from Eq. 共11兲关Fig. 2共a兲兴. From basic mechan-

ics the vertical shear stress, vd,i, is taken equal to the horizontal

shear stress 共xy =yx兲mobilized along the interface for a given

slip magnitude, si.

Deformation Estimates at Yield and Ultimate

The cross section is considered to have attained a state of ﬂexural

yielding when the extreme layer of tensile reinforcement reaches

ﬁrst its yield strain 共sy兲or alternatively when the concrete

strain at the extreme compression ﬁber exceeds the limit value of

c=1.5% 共FIB 2003, Chap. 4兲. Deﬁnition of an ultimate state is

also adopted so as to allow comparisons between the monolithic

and the detailed analytical approach. To this purpose an equiva-

lent monolithic curvature,u,M

eq , is estimated from the analysis

corresponding to a speciﬁed target drift at ultimate. The total

inelastic displacement comprises the elastic displacement at yield

⌬y, the plastic displacement ⌬p,u, and the displacement owing to

interface slip ⌬slip,u

⌬o=⌬y+⌬p,u+⌬slip,u共13a兲

where

⌬y=1

3yLs

2;⌬p,u=共u−y兲lp共Ls− 0.5lp兲;

⌬slip,u=slip,uLs=共s1,u+s2,u兲Ls

jd

共13b兲

In Eq. 共13b兲,y⫽curvature at yield of the composite section;

lp⫽length of the plastic hinge region 共taken here as 0.08Ls

+0.022Dbfsy according to Paulay and Priestley 1992兲; and

u⫽curvature at ultimate. Terms in Eq. 共13兲are calculated using

the proposed model and represent the tip displacements of a can-

tilever having a length Lsequal to the shear span of the member

共in seismic loading the cantilever considered represents approxi-

mately half the member length under lateral sway兲.⌬slip,uis

calculated at the ultimate from the slip values at the upper and

bottom interfaces, s1uand s2u, as shown in Fig. 2共d兲. Owing to

interface slip the cross section rotates by slip,u=共s1,u+s2,u兲/jd,

where jdrepresents the distance between the upper and the bot-

tom interfaces 共i.e., jdequals the core height which is usually the

cross-sectional height of the old member兲. Clearly, the end 共slip兲

rotation slip,uis greater for smoother interfaces, and therefore

deformation indices of jacketed members are expected to be

higher for lower interface friction properties.

The equivalent monolithic curvature, u,M

eq , is obtained by as-

suming equal displacements at ultimate for both the monolithic

and the composite members. Therefore

u,M

eq =

关

⌬o−1

3y,MLs

2+y,Mlp共Ls− 0.5lp兲

兴

lp共Ls− 0.5lp兲共14兲

In Eq. 共14兲,⌬o⫽total tip displacement 关calculated from

Eq. 共13a兲兴; and y,M⫽curvature at yield of the monolithic cross

section obtained from conventional sectional analysis.

Calculation Algorithm

In the proposed model the interfaces between old concrete and the

jacket are treated as the weak link of the composite behavior;

thus, the shear force demand introduced in the contact surfaces

for any level of ﬂexural curvature cannot exceed the associated

interface strength that corresponds to the level of slip already

attained. Calculations are performed for monotonically increasing

curvature using stepwise iteration. Initially, interface slip is taken

to be zero at both contact surfaces. Hence, in the ﬁrst step of the

solution 共for very small strains兲the longitudinal strain proﬁle is

identical to that of the monolithic approach 关Fig. 1共a兲兴. The shear

ﬂow is calculated from dual-section analysis, using the estimated

ﬂexural stresses. Based on classical mechanics a longitudinal

shear ﬂow 关i.e., shear stress xy in Fig. 1共d兲兴 may be calculated at

any distance yi关Fig. 1共a兲兴 from the neutral axis of a monolithic

elastic cross section, as

qo=VSi/I共15兲

where Si⫽ﬁrst moment of area from yito the top of the cross

section; I⫽moment of inertia of the composite cross section; and

V⫽shear force on the member calculated from the estimated ﬂex-

ural moment of the monolithic section divided by the shear span,

Ls. In the subsequent steps the longitudinal strain gradient is

modiﬁed 共by allowing for sequentially increasing discontinuities

of strain at the interface levels兲as required to satisfy equilibrium.

The interface slip is related to the magnitude of strain discontinu-

ity at the upper and bottom interfaces, ⌬1, and ⌬2, as follows:

s1=⌬1c=共c1−c2兲c,s2=⌬2c=共j3−c2兲c共16兲

where variables c1,j2,j3, and c2⫽normal strains in the section

layers above and below the contact surfaces 关Fig. 1共a兲兴; and

c⫽average crack spacing 关Fig. 2共a兲兴. Interface shear resistance is

mobilized depending on the slip magnitude: interface shear resis-

tances v1and v2关Eq. 共1兲兴 are obtained from the respective slip

values 共s1and s2兲using the constitutive relationships for interface

behavior 关Eqs. 共3兲–共8兲, illustrated in Fig. 2共c兲兴. Shear demand

values 共vd,1 and vd,2兲, estimated from Eq. 共12兲, are compared with

the dependable resistance values 关from Eq. 共1兲兴 for equilibrium. If

equilibrium is not attained then the slip estimate is subsequently

revised and the above calculation repeated until convergence. The

ﬁnal step in the algorithm involves establishing equilibrium of

forces over the composite member cross section. The strain pro-

ﬁle of the cross section is revised if there is a nonzero residual

section force resultant, i.e., if ⌺Fi−Next ⫽0; the algorithm con-

verges to a ﬁnal solution when both equilibrium requirements are

satisﬁed.

The algorithm is summarized in the ﬂowchart presented in

Fig. 3. It comprises the following steps:

1. For a selected level of sectional curvature, n, it is required

to calculate the associated moment resultant, Mn.共Note that

1376 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007

problem unknowns are: the normal strain at the top ﬁber of

the jacketed cross section, J1

n,m; the interface slip at the upper

共s1

n,r兲and bottom interfaces 共s2

n,r兲; and the associated moment

resultant, Mn兲. Therefore, start by setting the sectional curva-

ture equal to n;

2. Estimate normal strain at the top ﬁber of the cross section,

J1

n,m关Fig. 1共a兲兴;

3. Estimate the interface slip at the upper and bottom interfaces,

s1

n,rand s2

n,r关Eq. 共16兲兴. Crack spacing is calculated from

Eq. 共11兲;

4. Calculate the shear stress at the upper and bottom interfaces,

v1

n,rand v2

n,r, from the respective slip values, s1

n,rand s2

n,r

关Eqs. 共3兲–共8兲兴;

5. Deﬁne shear stress demands, vd,1

n,rand vd,2

n,r关Eq. 共12兲兴.If

both v1

n,r=vd,1

n,rand v2

n,r=vd,2

n,rproceed to Step 6, otherwise

return to Step 3 and set s1

n,r+1 =s1

n,r+ds1,s2

n,r+1 =s2

n,r+ds2.dsiis

the selected increment in the slip value;

6. Check cross-section equilibrium. If ⌺Fi−Next艋tolerance

go to Step 7. In any other case return to Step 2 and set

J1

n,m+1 =J1

n,m+dJ.dJis the step increment in the top strain

of the jacketed cross section;

7. Set J1

n=J1

n,m,s1

n=s1

n,r,s2

n=s2

n,rand store convergent values;

and

8. Estimate the moment resultant Mn. Repeat Steps 1–7 for

n=n+1. Calculations stop when the capacity of the shear

interface is exhausted.

Experimental Validation

Although RC jacketing is one of the most commonly applied

rehabilitation methods worldwide, a limited number of experi-

mental programs on RC jacketed subassemblages have been re-

ported 共Ersoy et al. 1993; Rodriguez and Park 1994; Bett et al.

1998; Gomes and Appleton 1998; Bousias et al. 2004; Vandoros

and Dritsos 2006a,b兲. The rather limited experimental database

关compared to ﬁber reinforced polymer 共FRP兲jacketing, for ex-

ample兴is a serious impediment in the development of design

expressions for this upgrading methodology.

In order to investigate the validity of the proposed analytical

model for RC jacketed members published experimental data

are used. From among the available tests those conducted by

Rodriguez and Park 共1994兲, Gomes and Appleton 共1998兲, Bousias

et al. 共2004兲, and Vandoros and Dritsos 共2006a,b兲summarized in

Table 1 are used for model veriﬁcation as they are considered

representative examples of columns under combined ﬂexure and

shear. It is noted that reinforcement slip owing to bond was

included. Other relevant studies that were not included either

concerned short column specimens 共Bett et al. 1998兲, or tests

that had been conducted under constant moment 共no shear, Ersoy

et al. 1993兲.

Details of the experimental program are outlined in Table 1 for

all specimens considered such as geometric properties and rein-

forcement details of the original as well as the jacketed elements.

In the identiﬁcation code adopted for the present comparative

study the ﬁrst character is either S or M corresponding to

strengthened members with jacketing after cyclic loading or

specimens built monolithically with a composite section to be

used as controls, respectively. The second character represents the

treatment at the interface: rcorresponds to roughened interface

achieved by chipping or sandblasting or other such methods,

whereas srepresents a smooth interface. The third character 共Dor

N兲identiﬁes specimens with dowels 共marked by D兲or without

dowels 共marked by N兲crossing the interface between the interior

core and the jacket. The fourth character pd corresponds to pre-

damaged units. The numerals 15, 25, 30, and 45 stand for the

lap splice length of the existing unit corresponding to 15Db,

25Db,30Db, and 45Db, respectively. The character lcorresponds

to U-shaped steel links utilized to connect the longitudinal rein-

forcement of the jacket to the existing member 共core兲and the

character wcorresponds to welding of stirrup ends of the ﬁrst four

stirrups 共from the base of the jacketed member兲. The numeral in

the end is the specimen number considered 共in successive order兲

in Table 1. For easy reference, the original code names used

for the specimens by the original investigators are also listed in

Table 1 共column “Specimens”兲.

Results

The calculated lateral load versus lateral displacement curves

along with the curves obtained from standard sectional analysis of

the monolithic cross sections for the total number of tested units

are plotted in Figs. 4–8. The experimental curves plotted on the

same ﬁgures represent the envelope of the recorded lateral load

versus lateral displacement hysteretic loops.

In general, the monolithic approach grossly overestimates the

actual response of the jacketed member; however it is successful

in reproducing the trends of member behavior even if interface

slip is neglected. The analytical model provides a lower bound of

the response of the jacketed members and it may generally be

considered conservative, while matching well the experimental

values. At low deformation levels response curves obtained by the

analytical approach and by the monolithic approach almost coin-

cide. This is expected as long as crack formation is at an early

stage.

In addition to these general observations, the following points

are noted: for the ﬁrst group of specimens 共Rodriguez and Park

Fig. 3. Flowchart of proposed algorithm

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1377

Table 1. Summary of Test Units

Group Code nameaSpecimenb

bc

共mm兲

hc

共mm兲

Db,c

共mm兲

lc

共%兲

Dbs,c

共mm兲

pwc

共%兲

fc

共MPa兲

fsy

共MPa兲

bj

共mm兲

hJ

共mm兲

Db,J

共mm兲

lJ

共%兲

Dbs,J

共mm兲

wJ

共%兲

fc

共MPa兲

fsy

共MPa兲Nc

Ls

共mm兲

1st

Rodriguez and Park

共1994兲

SrNpd-1 SSI 350 350 20 2.05 6 0.16 29.5 325 550 550 16 0.89 10 0.36 32.9 502 10.0 1,425

SrN-2 SS2 350 350 20 2.05 6 0.16 29.5 325 550 550 16 0.89 10 0.36 34.0 502 10.0 1,425

SrN-3 SS3 350 350 20 2.05 6 0.16 29.5 325 550 550 12 0.75 10 0.94 19.4 491 10.0 1,425

SrNpd-4 SS4 350 350 20 2.05 6 0.16 25.9 325 550 550 12 0.75 10 0.94 25.2 491 10.0 1,425

2nd

Gomes and Appleton

共1998兲

SsNpd-5 P2R 200 200 12 1.13 6 0.22 53.2 480 260 260 12 1.64 6 0.33 58.2 480 6.0 1,000

SsNpd-6 P3R 200 200 12 1.13 6 0.66 58.2 480 260 260 12 1.64 6 0.49 49.6 480 7.1 1,000

MsN-7 P4 200 200 12 1.13 6 0.22 56.2 480 260 260 12 1.64 6 0.33 56.2 480 6.3 1,000

3rd

Bousias et al.

共2004兲

MsN-8 Q-RCL0M 250 250 14 0.98 8 0.24 30.6 313 400 400 20 1.29 10 0.44 30.6 500 18.0 1,600

SsN-9 Q-RCL0 250 250 14 0.98 8 0.24 26.3 313 400 400 20 1.29 10 0.44 55.8 500 7.9 1,600

SsN15-10 Q-RCL1 250 250 14 0.98 8 0.24 27.5 313 400 400 20 1.29 10 0.44 55.8 500 8.4 1,600

SsN25-11 Q-RCL2 250 250 14 0.98 8 0.24 25.6 313 400 400 20 1.29 10 0.44 55.8 500 8.4 1,600

SsNpd15-12 Q-RCL01pd 250 250 14 0.98 8 0.24 28.1 313 400 400 20 1.29 10 0.44 20.7 500 25.0 1,600

SsNpd25-13 Q-RCL02pd 250 250 14 0.98 8 0.24 28.6 313 400 400 20 1.29 10 0.44 20.7 500 27.0 1,600

4th

Bousias et al.

共2004兲

SsN15-14 R-RCL1 250 500 18 0.81 8 0.24 36.7 514 400 650 18 1.13 10 0.44 55.8 500 6.6 1,600

SsN30-15 R-RCL3 250 500 18 0.81 8 0.24 36.8 514 400 650 18 1.13 10 0.44 55.8 500 6.6 1,600

SsN45-16 R-RCL4 250 500 18 0.81 8 0.24 36.3 514 400 650 18 1.13 10 0.44 55.8 500 5.2 1,600

5th

Vandoros and Dritsos

共2006a,b兲

MsN-17 Q-RCM 250 250 14 0.98 8 0.24 24.7 313 400 400 20 1.29 10 0.44 24.7 487 21.2 1,600

SsNl-18 Q-RCW 250 250 14 0.98 8 0.24 22.9 313 400 400 20 1.29 10 0.44 18.8 487 21.6 1,600

SsD-19 Q-RCD 250 250 14 0.98 8 0.24 27.0 313 400 400 20 1.29 10 0.44 55.8 487 8.9 1,600

SrN-20 Q-RCR 250 250 14 0.98 8 0.24 27.0 313 400 400 20 1.29 10 0.44 55.8 487 8.9 1,600

SrD-21 Q-RCRD 250 250 14 0.98 8 0.24 27.0 313 400 400 20 1.29 10 0.44 55.8 487 8.9 1,600

SsNw-22 Q-RCNT 250 250 14 0.98 8 0.24 27.0 313 400 400 20 1.29 10 0.44 17.8 487 25.6 1,600

SsN-23dQ-RCNTA 250 250 14 0.98 8 0.24 23.8 313 400 400 20 1.29 10 0.44 34.5 487 11.8 1,600

SsDw-24 Q-RCE 250 250 14 0.98 8 0.24 36.8 313 400 400 20 1.29 10 0.44 24.0 487 20.6 1,600

aS⫽strengthened members with jacketing, M⫽specimens built monolithically, r⫽roughened interface, s⫽smooth interface, D, N⫽specimens with or without dowels, respectively, pd⫽predamaged units,

15, 25, 30, and 45: stand for the lap splice length corresponding to 15Db,25Db,30Dband 45Db;b⫽U-shaped steel links utilized, w⫽welding of stirrup ends of the ﬁrst four stirrups, the numeral in the

end is the specimen number considered 共in successive order兲.

bOriginal code names used for the specimens by the original investigators.

cAxial load ratio % calculated on the basis of concrete strength of the jacket.

dJacket constructed under axial load.

1378 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007

1994, Fig. 4兲the previous damage of units SrNpd-1 and SrNpd-4

had no signiﬁcant inﬂuence on the response as compared to units

SrN-2 and SrN-3, which had not suffered any damage prior to

jacketing. Clearly, the analytical result is very close to its experi-

mental counterpart in the case of specimens MsN-7 and SsNpd-5

共Gomes and Appleton 1998, Fig. 5兲. The experimental curve rep-

resenting specimen SsNpd-5 lies below that of specimen MsN-7

and this is attributed to the initial damage of unit SsNpd-5. In the

third group 共Bousias et al. 2004, Fig. 6兲the experimental response

of the units seems insensitive to the lap splice length of existing

reinforcement and to the degree of previous damage imparted to

units SsNpd15-12 and SsNpd25-13. This is also observed in the

case of the fourth group of units 共Bousias et al. 2004, Fig. 7兲.In

the last group of units 共Vandoros and Dritsos 2006a,b Fig. 8兲the

estimated strength of the monolithic unit 共MsN-17兲matches the

experimental evidence but the actual secant to yield stiffness is

lower. The response of unit SsNw-22 is very close to the response

of the monolithic approach, although slip at the interface modiﬁes

the response somewhat, as shown by the analytical curve.

The response of the jacketed members is inﬂuenced greatly by

the interface model utilized. A more sensitive model that could

describe in more detail the interface shear behavior would provide

better results. In general, a softer response than the experimental

envelope implies too compliant an interface, whereas the opposite

trend implies the interface stiffness has been overestimated. This

is demonstrated in the following sections, where a parametric

investigation of the model’s sensitivity is explored. Interface be-

havior requires further calibration, and this would have been done

if a critical mass of experiments were available. However, even as

things stand, by explicitly accounting for this aspect in calculating

the ﬂexural response of composite members, the model introduces

a degree of freedom that enables consideration of an important

response mechanism that was previously overlooked.

Parametric Investigation

A parametric investigation is conducted in the present section so

as to establish the sensitivity of the monolithic factors to the

important design and model variables. Note that these factors

are used to estimate the response indices of jacketed, composite

reinforced concrete members, from the corresponding response

variables of monolithic members with identical cross section, on

the premise that the latter quantities are easily established from

conventional ﬂexural analysis. The magnitude of monolithic fac-

tors depends on the property considered 共strength, stiffness, or

deformation兲, on the jacket characteristics and on the interface

properties.

Parameters of Study

A sensitivity analysis of monolithic factors is conducted in this

section through a detailed evaluation of two reference cases. The

core of the composite member is the existing member, represen-

tative of former construction practices. In Case 1 the core used

had a 350 mm square cross section, reinforced longitudinally

with a steel area ratio, lc, equal to 1% and transverse conﬁning

reinforcement ratio wc =0.13% 共perimeter stirrups Ø6 / 200 mm兲.

Concrete cylinder uniaxial compressive strength fc

⬘was 16 MPa

and steel yield strength fsy was 300 MPa. In Case 2 the core had

a rectangular cross section of 250 by 500 mm, with a longitudinal

reinforcement ratio, lc =0.8%, transverse conﬁning reinforcement

ratio wc =0.24% 共perimeter stirrups Ø8 mm/ 200 mm兲, concrete

uniaxial compressive strength fc

⬘=16 MPa, and steel yield

strength fsy =300 MPa. In both cases the jacket considered was

Fig. 4. Lateral load versus drift for ﬁrst group of units 共adapted from

Rodriguez and Park 1994兲

Fig. 5. Lateral load versus drift for second group of units 共adapted

from Gomes and Appleton 1998兲

Fig. 6. Lateral load versus drift for third group of units 共adapted from

Bousias et al. 2004兲

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1379

75 mm thick. After application of the jacket the shear span ratio

was reduced from 4.3 to 3 共ﬂexure dominated兲and from 3 to 2.3

共shear dominated兲for the two case studies, respectively.

Parameters of the investigation were the percentage of

the longitudinal reinforcement of the jacketed cross section

关lJ =AJ/共bJhJ−bchc兲兴 which varied between 1 and 3%, the trans-

verse conﬁning reinforcement ratio of the jacket 共wJ兲which var-

ied for the square cross section 共Case 1兲between 0.3 and 1.25%,

and for the rectangular cross section 共Case 2兲between 0.4 and

1.75% and the axial load 共N兲applied on the jacketed cross section

expressed as a fraction of the theoretical crushing capacity 共Agfc

⬘兲

of the jacketed cross section which varied between 0 and 0.3. The

cylinder compressive strength of the jacket concrete was taken as

fc

⬘=20 MPa. Yield strength of both longitudinal and transverse

jacket reinforcements was taken as fsy = 500 MPa.

The results of the parametric study are presented in terms of

the monolithic factor values both for ﬂexural strength and for

deformation capacities. In this regard, the following three deﬁni-

tions are adopted for the objectives of the study

Ky

M=My

My,M

;Ku

M=Mu

Mu,M

共17a兲

Ky

=y

y,M

;Ku

=u

u,M

共17b兲

K,=

,M

;K⌬,=⌬

⌬,M

共17c兲

where KM,K, and K⫽monolithic factors for ﬂexural strength,

curvature, and ductility. Subscripts yand u⫽yield and ultimate,

respectively; whereas and ⌬⫽curvature and displacement duc-

tilities. The moments at yield, My, and ultimate, Mu, of the RC

jacketed member are estimated by multiplying the corresponding

moments, My,M, and Mu,M, of the monolithic member with factors

Ky

Mand Ku

M关Eq. 共17a兲兴. Pertinent monolithic factors Ky

and Ku

may be used in the same way in order to obtain the curvature

at yield, y, and ultimate, u, of the RC jacketed members

关Eq. 共17b兲兴. Similarly, by multiplying the curvature ductility ,M

and the displacement ductility ⌬,M, of the monolithic cross sec-

tion with appropriate monolithic factors K,and K⌬,, the curva-

ture ductility and the displacement ductility ⌬of the jacketed

member may be estimated 关Eq. 共17c兲兴.

Role of Characteristics of Jacket

The direct effect induced by any change in the design character-

istics of the jacket is depicted for both the yield and the ultimate

stage in Fig. 9. The circular mark in Fig. 9 corresponds to the

reference case of the parametric study with lJ =1%, wJ= 0.3%,

and N⬘=0 for the square section example, and lJ =1%,

wJ =0.4%, and N⬘= 0 for the rectangular one. The arrows indicate

the inﬂuence on the monolithic factors plotted in the xand yaxes,

effected by a corresponding change in the parameter studied.

With reference to the square cross section 共Case 1兲, increasing

both the percentage of longitudinal reinforcement of the jacket

共lJ and the applied axial load ratio 共N⬘=N/Agfc

⬘兲results in a

reduction of Ky

Mand an increase of Ky

关Fig. 9共a兲兴. This is also

observed in the shear dominated member 共Case 2兲, however, the

inﬂuence is less pronounced, especially on Ky

M. This indicates that

ﬂexure-dominated members are more sensitive to changes of

axial load and longitudinal reinforcement compared to the shear-

dominated ones. As discussed earlier, the jacketed member

reaches a yield at lower strength but at increased curvature as

compared to its monolithic counterpart, owing to the increased

Fig. 7. Lateral load versus drift for fourth group of units 共adapted from Bousias et al. 2004兲

Fig. 8. Lateral load versus drift for ﬁfth group of units 共adapted from

Vandoros and Dritsos 2006a,b兲

1380 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007

deformation due to interface slip. The opposite is observed when

conﬁnement reinforcement of the jacket 共wJ兲is increased since

interface slip is suppressed with conﬁnement 共i.e., the cross sec-

tion approaches more toward the monolithic condition兲.

Results of the parametric investigation at a nominal ultimate

limit state for both reference Cases 1 and 2 are presented in

Fig. 9共b兲. The nominal ultimate is taken here to correspond to a

lateral drift of 2% for both the analytical and monolithic model.

This level was selected as a performance limit state and a point of

reference as it corresponds to a displacement ductility in excess of

3 for regular frame members, which is considered an upper bound

for the acceptable level of ductility demand in a redesigned struc-

ture. Increasing the longitudinal jacket reinforcement ratio 共lJ兲

and applied axial load ratio 共N⬘兲produce a simultaneous reduc-

tion in the monolithic factors for strength and deformation at

ultimate, whereas the reverse effect is obtained by increasing the

amount of jacket conﬁnement reinforcement 共wJ兲关Fig. 9共b兲for

Case 1兴. In the case of the shear-dominated member 共Case 2兲the

response is differentiated with regards to the axial load inﬂuence:

as the axial load ratio increases the monolithic factor for strength

at ultimate also increases, whereas the monolithic factor for de-

formation at ultimate decreases.

The inﬂuence that each of the parameters under investigation

has on the various monolithic factors is depicted in Figs. 10–12

and is discussed in detail in the following subsections for both

the square 共Case 1兲and rectangular 共Case 2兲cross sections,

respectively.

Longitudinal Jacket Reinforcement Ratio „lJ…

Ky

M,Ku

M,Ku

,K,, and K⌬,are all reduced with increasing value

of this variable 共Fig. 10兲. The reverse trend is observed for Ky

.

Axial Load „N⬘…

Increasing the applied axial load ratio 共N⬘兲leads to a simulta-

neous reduction of Ky

M,Ku

M, and Ku

共Fig. 11兲, but also of K,and

K⌬,. The monolithic factor of curvature at yield 共Ky

兲increases

for an axial load ratio up to 0.2, but the trend is not uniform for

both cases 共Case 1 and 2兲at higher axial loads.

Conﬁning Reinforcement „wJ…

As illustrated in Fig. 12, Ky

Mand Ku

Mincrease mildly as the per-

centage of jacket conﬁning reinforcement 共wJ兲increases. Be-

cause the dowel function of transverse reinforcement is mobilized

passively, Ky

is almost insensitive to wJ, whereas there is a

strong increase of Ky

with conﬁnement. Similarly, K,and K⌬,

both increase with wJ.

Fig. 9. Monolithic factors of strength and deformation at: 共a兲yield;

共b兲ultimate

Fig. 10. Inﬂuence of jacket longitudinal reinforcement on monolithic

factors 共Case 1, Case 2兲

Fig. 11. Inﬂuence of axial load on monolithic factors 共Case 1,

Case 2兲

Fig. 12. Inﬂuence of jacket conﬁnement reinforcement on monolithic

factors 共Case 1, Case 2兲

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1381

Discussion of Results of Parametric Study

The results of the current parametric study provide an insight

into the mechanical effect that jacket characteristics play on

the lateral load response of jacketed members. Monolithic factors

are sensitive to the design variables of the jacket and do

not generally assume an obvious ﬁxed value. From the results of

the parametric study for both the square 共Case 1兲and the rectan-

gular 共Case 2兲cross section the values of the monolithic factors

range as follows: 共1兲Ky

M=共0.63–0.95兲;共2兲Ku

M=共0.49–0.97兲;共3兲

Ky

=共0.95–2.57兲;共4兲Ku

=共0.34–0.90兲;共5兲K,=共0.15– 0.93兲;

and 共6兲K⌬,=共0.39–0.94兲.

The above results are consistent with the values suggested by

EC8 共CEN 1996兲for the monolithic factor of strength KR=0.8 共no

differentiation is made by the code between yield and ultimate兲,

although the range of estimated values is larger for the ultimate

共Ku

M兲. The estimated values for Ky

show that jacketed cross

sections reach yield at greater curvatures, owing to slip at the

interface between the existing member and the core. The Ku

is

less than 1.0, thus, in general the curvature at ultimate 共u兲esti-

mated from the analytical approach is smaller than the monolithic

estimate 共u,M兲. Considering that slip at the upper and bottom

interfaces contributes to lateral drift, the reduced value of curva-

ture at ultimate 共2% drift兲deﬁned by the analytical approach is

justiﬁed. The monolithic factors of curvature and displacement

ductilities 共K,,K⌬,兲are less than 1.0, which emphasizes that

analytical curvature and displacement ductilities are both lower

than the corresponding monolithic values.

Sensitivity of Analytical Model

The proposed analytical model is primarily sensitive to param-

eters that affect the estimation of crack spacing and the shear

strength of the contact interfaces. Each of these variables has a

distinct inﬂuence on the computational procedure; however,

selection of the shear interface model is fundamental. Variables of

the shear transfer model used herein 共Tassios and Vintzēleou

1987; Vintzēleou and Tassios 1986, 1987; Vassilopoulou and

Tassios 2003兲are the interface strength 共vc,u兲and the slope of the

postpeak branch 共兲关Fig. 13共a兲兴.

A brief parametric investigation was conducted in order to

explore the sensitivity of the model to the primary variables.

The square cross section used in the preceding as Case 1 is

used as a point of reference. Geometric characteristics and mate-

rial properties of the existing member core were already given

in earlier sections. Longitudinal jacket reinforcement ratio

was selected as lJ =1%, with transverse conﬁning reinforce-

ment wJ =0.3% 共fsy =500 MPa兲. No axial load was applied on

the jacketed cross section, whereas fc

⬘=20 MPa for the jacket

concrete.

First, the inﬂuence of the interface shear friction on the re-

sponse of the jacketed member was studied. To model unfavor-

able conditions at the interface, the shear friction coefﬁcient is

increased stepwise up to 0.65 共values used are: 0.4, 0.55, 0.65

while keeping =1; this coefﬁcient indirectly accounts for the

roughness of the interface兲. The steepness of the descending

branch was examined for =1, 1 / 2, and 1 / 3, whereas vc,u

was given by Eq. 共4兲. The values selected for are based on

published experimental data 共Bass et al. 1989; Papanicolaou and

Triantaﬁllou 2002兲. The results of the parametric investigation

are summarized in Fig. 13 in a moment versus curvature diagram.

For lower values of , i.e., more gradual decay of the descending

branch of the shear stress strain curve 关Fig. 13共a兲兴, higher levels

of curvature capacity are estimated 关Fig. 13共b兲兴. Increasing

leads to higher shear capacity at the contact surface allowing

for the development of higher strength and curvature values

关Fig. 13共c兲兴.

Summary and Conclusions

An algorithm for calculating the monotonic response of rein-

forced concrete jacketed members is presented. The model intro-

duces a kinematic degree of freedom 共interface slip兲that enables

consideration of an important mechanism of behavior that was

previously overlooked, namely the shear transfer mechanisms

mobilized due to sliding at the interface between existing and new

material. The weak link controlling deformations in this problem

is the interface. The capacity of the weakest link is evaluated and

checked in every step, to make sure it is not exceeded by the

demand. The shear demand at the interface is controlled by the

ﬂexural stresses on the cross section and by the spacing of cracks

in the longitudinal direction, whereas the shear capacity is a func-

tion of slip. The shear stress slip relationship for the contact sur-

faces and the deﬁnition of crack spacing play a key role in the

algorithm. Analytical results show that the model can reproduce

successfully the observed response of jacketed members and cor-

relates well with experimental data. This analysis tool was used to

explore the difference between the ideal response of monolithic

members and the actual response of the RC jacketed members of

identical geometry with reference to the design variables. A para-

metric study was conducted and the dependence of various mono-

lithic factors on the characteristics of the jacket was investigated.

It was found that strength factors at yield 共Ky

M兲range between

0.63 and 0.95, whereas strength factors at ultimate 共Ku

M兲range

between 0.49 and 0.97. Monolithic factors for deformation indi-

ces were found in the case of curvature at yield to range between

0.95 and 2.57, whereas in the case of curvature at ultimate be-

tween 0.34 and 0.90. The monolithic factors of curvature and

displacement ductilities 共K,,K⌬,兲are both lower than the cor-

responding monolithic values with the former to range between

0.15 and 0.93 and the latter between 0.39 and 0.94.

Fig. 13. Role of shear friction interface model

1382 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007

Acknowledgments

The ﬁrst writer was co-funded by the European Research Project

“Seismic Performance Assessment and Rehabilitation” 共SPEAR兲,

through Imperial College of Science, Technology and Medicine

共London, UK兲and by the Hellenic Ministry of Education and

Religion Affairs through the scholarship “HRAKLEITOS.” The

contribution of the second writer was funded by the Hellenic

Secretariat of Research and Technology 共GSRT兲through the

multi-Institutional Project “ARISTION.” The contribution of the

third author was funded by the US National Science Foundation

through the Mid-America Earthquake Center, Award No. EEC

97-01785.

Notation

The following symbols are used in this paper:

Ag⫽gross section area;

Aint ⫽contact area of interface;

bJ⫽width of jacketed cross section;

c⫽average crack spacing;

Db⫽diameter of bars clamping interface;

Db,c,DbJ ⫽bar diameter of core and jacket longitudinal

reinforcement, respectively;

Es⫽elastic modulus of steel;

Ecm ⫽elastic modulus of concrete;

fb,c,fb,J⫽average bond stress of core and jacket

reinforcement layer, respectively;

fc

⬘⫽concrete cylinder uniaxial compressive

strength;

fctm ⫽tensile strength of concrete;

fctm,c⫽tensile strength of concrete core;

fs⫽axial stress of bars crossing interface;

fs,cr ⫽jacket steel stress at crack;

fsy ⫽yield strength of steel;

I⫽moment of inertia of composite cross

section;

jd⫽distance between upper and bottom

interfaces;

Ki⫽monolithic factor, where subscript i=R,K

refers to strength and stiffness, respectively;

KM,K,K⫽monolithic factors for ﬂexural strength,

curvature, and ductility, respectively;

k⫽number of dowels;

Ls⫽shear span of member;

lc⫽height of tension zone in core component of

composite cross section;

lp⫽length of plastic hinge region;

M⫽moment resultant;

N⫽axial load;

N⬘共=N/Agfc

⬘兲⫽applied axial load ratio;

Nc,NJ⫽number of bars in tension steel layer of core

and jacket, respectively;

Next ⫽externally applied axial load;

p⫽normal pressure externally applied on

contact plane;

qs⫽shear ﬂow at contact between bottom layer

and core;

q0⫽shear ﬂow based on classical mechanics;

Si⫽ﬁrst moment of area;

s⫽lateral slip;

sc,u⫽ultimate slip value beyond which frictional

mechanisms break down;

Sd,el ⫽elastic slip value;

Sd,u⫽ultimate slip value;

s1u,s2u⫽slip values at upper and bottom interfaces of

jacket;

V⫽shear force on member;

VD共s兲⫽dowel force estimated for slip magnitude, s;

VD,u⫽ultimate dowel force;

va⫽shear resistance of aggregate interlock

mechanism;

vc共s兲⫽frictional resistance at slip, s;

vc,u⫽ultimate frictional resistance of interface;

vD⫽shear stress resisted by dowel action in

cracked reinforced concrete;

vd,1,vd,2 ⫽shear demand values;

vn⫽shear resistance;

v1,v2⫽shear resistances at upper and bottom

interfaces;

w⫽separation between contact surfaces as they

slide overriding one another;

␣⫽bar axial stress normalized with respect to

its yield value;

⌬p,u⫽plastic displacement;

⌬slip,u⫽displacement owing to interface slip;

⌬y⫽elastic displacement at yield;

⌬0⫽total tip displacement;

⌬1,⌬2⫽magnitude of strain discontinuity at upper

and bottom interfaces, respectively;

c⫽concrete strain at extreme compression ﬁber;

c1,j2,j3,c2⫽normal strains in section layers above and

below contact surfaces;

sy ⫽tensile reinforcement yield strain;

共=Es/Ecm兲⫽modular ratio;

slip,u⫽rotation owing to interface slip;

⫽coefﬁcient indirectly accounting for

roughness of interface;

⫽interface shear friction coefﬁcient;

⌬⫽displacement ductility;

⫽curvature ductility;

lc ⫽percentage of longitudinal reinforcement of

existing cross section 共core兲;

lJ ⫽percentage of longitudinal reinforcement of

jacketed cross section;

s⫽reinforcement area ratio;

s,eff ⫽effective reinforcement ratio;

wc ⫽transverse conﬁning reinforcement ratio of

existing cross section 共core兲;

wJ ⫽transverse conﬁning reinforcement ratio of

jacketed cross section;

⌺Fi⫽normal stress resultant of each layer, i;

N⫽normal clamping stress acting on interface;

xy ⫽shear stress acting on plane with unit normal

parallel to x-axis, and oriented in y-axis

共deﬁned according to classical mechanics as

per Fig. 2兲;

xz ⫽shear stress acting on plane with unit normal

parallel to x-axis, and oriented in z-axis,

respectively 关Fig. 1共d兲兴;

⫽curvature;

u⫽curvature at ultimate;

u,M

eq ⫽equivalent monolithic curvature;

y⫽curvature at yield of composite section; and

y,M⫽curvature at yield of monolithic cross

section.

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / OCTOBER 2007 / 1383

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