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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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