FRPRCS-8
University of Patras, Patras, Greece, July 16-18, 2007
DIAGONAL STRESS-STRAIN MODEL FOR FRP-CONFINED
RECTANGULAR CONCRETE COLUMNS
Domingo A. MORAN 1
1
2
Chris P. PANTELIDES 2
Ph.D. Candidate, University of Utah, Salt Lake City, Utah, USA
Professor, University of Utah, Salt Lake City, Utah, USA
Keywords: concrete columns; composite materials; ductility; rehabilitation; retrofit; seismic hazard;
stress-strain relations.
1 INTRODUCTION
Despite the development of the ACI440.2R-02 [1] design guidelines and the successful
application of Fiber Reinforced Polymer (FRP) jacketing systems for circular, square and
rectangular columns, research into the constitutive relationships governing the dilation and
compressive behavior of rectangular FRP-confined concrete is continuing [2,3,4,5].
For circular FRP-confined concrete sections subjected to uniform axial compression the
confining FRP jacket provides a uniform kinematic restraint at the surface of the concrete core; and
subjects the concrete to a uniform triaxial compressive state of stress. However, for rectangular
and square FRP-confined concrete sections, the effectively confined concrete core is in a complex
triaxial compressive state of stress. The thin rectangular FRP jacket provides a high axial
kinematic restraint along the faces of the rectangular section as a result of its high axial stiffness,
and a very low out-of-plane kinematic restraint as a result of its low flexural stiffness. As a result of
the geometry of the rectangular FRP jacket, and the difference between the axial and flexural
stiffness of the jacket, the concrete core is subjected to a non-uniform triaxial compressive state of
stress. The resultant passive confining pressure provided by the FRP jacket decreases
significantly as the aspect ratio of the rectangular FRP-confined (FCC) or concrete filled FRP tube
(CFFT) section increases, as it is concentrated along the diagonal axis of the rectangular FCC or
CFFT section. Despite the development of stress-strain models for circular and square FRPconfined concrete sections there is a lack of models that can accurately predict the dilation and
compressive behavior of rectangular FRP-confined concrete sections of varying aspect ratios and
jacket corner radii.
In this paper, a new and unique stress-strain model for FRP-confined concrete sections is
introduced in which the concept of diagonal dilation and equilibrium of the FRP-confined concrete
is considered. A variable Poisson’s ratio model, developed by the authors [6], in combination with
analytical strain compatibility and diagonal equilibrium relationships developed herein for
rectangular FRP-confined concrete sections is combined with a Popovics [7] type uniaxial stressstrain model for FRP-confined concrete sections, using the following concepts: (1) The increase in
compressive strength and ductility of FRP-confined concrete are expressed in terms of the internal
damage resulting from diagonal dilation of concrete and lateral restraint provided by the FRP
jacket; (2) The dilation behavior of FRP-confined concrete sections in compression depends on the
mechanical properties of the unconfined concrete, the lateral restraint provided by the FRP jacket,
the geometry of the concrete section, and the extent of internal damage in the concrete core.
Modeling of both the compressive and dilation behavior of FRP-confined sections is achieved using
a minimal number of experimentally obtained coefficients, which were found to affect both the
compressive and dilation behavior.
2 CONFINEMENT EFFECTIVENESS OF FRP-CONFINED CONCRETE
The confinement effectiveness of the confining element
k cc , is measured by how much the
strength, f cc , and strain, ε cc , of the confined concrete is increased over the unconfined concrete
strength, f co and strain ε co using the following empirical relationships [8]:
1
ε
f
f
k cc = cc = 1 + k1k r ; kεc = cc = 1 + k 2 k r ; k r = r
f co
ε co
f co
(1)
where k1 and k 2 are experimentally determined confinement and strain effectiveness coefficients,
respectively; k εc = strain effectiveness; k r = confinement ratio; and f r = average confining
pressure.
Various investigators have proposed nonlinear-failure criteria for concrete, most of which are
empirical in nature. Considering that the structure of concrete material within the confined concrete
core is heterogeneous, consisting of granular aggregates (i.e. crushed stone or gravel and sand), a
binding material (i.e. cement paste) and pores. Treating the concrete core as a Mohr-Coulomb
frictional cohesive material with some inter-particle attraction or apparent cohesion cc ; the
concrete’s resistance to deformation is developed from a combination of particle sliding, rolling,
crushing and void compaction that develops during the deformation of the concrete core, which
depend on the average angle of internal friction φc of the confined concrete.
Fig. 1 Mohr-Coulomb failure criteria for frictional-cohesive materials (a) Mohr’s failure stress
circles including terminology (b) Mohr-Coulomb failure envelope versus normal stress.
τ c , of the concrete material as shown in Fig. 1, can be approximated by:
(σ − σ 3 )
φ
(2)
τc = 1
cos(2θ c ) = cc + σ n tan φ c ; θ c = 45° + c
The shear strength,
2
2
σn =
where
σ1 =
σ1 + σ 3
2
+
σ1 − σ 3
2
major principal compressive stress;
cos(2θ c )
σ 3 = minor
(3)
principal compressive stress; σ n
=
θ c = average angle of inclination of the failure shear
plane.
For the cases of concrete in uniaxial compression for which σ 3 = 0 and σ 1 = f cc = f co , and
normal stress on the failure shear plane;
σ 1 = f cc = 0 , and solving for the major
compressive stress σ 1 using Eqs. (2) and (3), setting σ 1 = f cc and cc = cu , yields:
uniaxial tension for which
σ 3 = − f to
⎛ 1 + sin φc
f cc = σ 3 ⎜⎜
⎝ 1 − sin φc
and
⎞
⎛ cos φc
⎟⎟ + 2cu ⎜⎜
⎠
⎝ 1 − sin φc
2
⎞
⎟⎟ = σ 3 tan 2 (θ c ) + 2cu tan (θ c )
⎠
principal
(4)
FRPRCS-8
University of Patras, Patras, Greece, July 16-18, 2007
cu =
f to
f
= co
2
2 kt
kt ; kt =
f to
; f to = 0.32( f co )2 3
f co
(5)
where cu is the apparent uniaxial cohesion intercept of the concrete core, as shown in Fig. 1;
k t = ratio of the uniaxial tensile strength f to to the uniaxial compressive strength f co of the
concrete core. Using Eqs. (4) and (5), setting k 3 = σ 3 f co , k 3e = k e k 3 , and setting k 3 = k 3e ,
the confinement effectiveness of the confined concrete core k cc of Eq. (1), can be found in terms
of its frictional-cohesive properties and the effective minor principal compressive stress ratio k 3e in
the rectangular FRP confined concrete section, as follows:
f
1 + sin φ c
k cc = cc = k t k1 + k 3e k1 ; k1 =
= tan 2 θ c
f co
1 − sin φ c
k 3e = α uψ 3 k de ; α u =
(6)
σ
1
; ψ 3 = 3e = [1 + cos(2θ d − 90°) + sin (2θ d − 90°)] (7)
f de 2
(α sh )2
1
α sh =
Rj
Hc ε H
=
= tan (θ d ) ; α j =
Bc
Hc
εB
(8)
⎛ 2t j k e ⎞
f
⎟E j
k de = de = E je ψ sh ε d = K jeγ sh ε d ; E je = ⎜⎜
⎟
B
B
f co
⎝ Hc ⎠
C je
2ψ sh
α sh
1
; γ sh =
; K je =
; C je = C sh E je
ψ sh =
B
C sh
f co
2
2 β d 1 − α jα sh
( )
( )
)
2
⎤
2 ⎡ (1 − 2α j α sh )
⎥
ke = 1 − ⎢
3 ⎢1 − α sh (4 − π )(α j )2 ⎥
⎦
⎣
( )
(
β d = β sh β eff ; β sh = 1 +
where
σ 3e =
α u = uniformity
1
αu
; C sh =
; β eff =
(1 + α sh ) − (4 − π )α jα sh
1 − α sh (4 − π )(α j )2
1
2 − eeff
; eeff = 1 − α u
(9)
(10)
(11)
(12)
effective minor principal compressive stress provided by the FRP jacket;
coefficient;
ψ3 =
minor principal stress coefficient;
compressive stress in the confined concrete core;
(E je )B =
σ3 =
minor principal
effective FRP jacket transverse
Bc of the section; ψ sh = diagonal equilibrium coefficient;
K je = normalized effective FRP jacket confinement stiffness; γ sh = jacket equilibrium
coefficient; k e = confinement efficiency of the FRP jacket; C je = effective FRP jacket
stiffness along the minor dimension
confinement stiffness;
α sh =
section aspect ratio;
αj =
jacket corner aspect ratio;
εd =
ε c ; C sh = jacket
β sh = aspect ratio
diagonal strain in the FRP-confined concrete section at any given axial strain,
confinement ratio coefficient;
βd =
diagonal strain compatibility coefficient;
β eff =
jacket coefficient; eeff = eccentricity of the effectively confined elliptical
concrete core. The effectively confined concrete core of the section of Fig. 2, is modeled as an
equivalent elliptical core having dimensions H eff , Beff and a distance between the foci of C eff ,
coefficient;
as shown in Fig 2(a). The concrete section is confined by an FRP jacket having a normalized
3
effective stiffness K je , subjected to an axial compressive strain
diagonal strain
εd ,
εc ,
along the diagonal axis D j , of the section.
inclination of the main diagonal D j ;
εB
and
εH
and a resultant expansive
In addition,
θd =
angle of
are the average FRP jacket strains along the
minor and major dimensions of the section, respectively.
Fig. 2 Typical Rectangular FRP-confined concrete section (a) Typical section geometry (b)
Diagonal equilibrium of section.
Note that the k t k1 and k 3e k1 terms of Eq. (6), can be considered to be the apparent cohesive
intercept parameter and apparent frictional slope parameter, respectively, of the confinement
effectiveness k cc , of confined concrete. Also, note that k cc of Eq. (6) yields k cc of Eq. (1) only
k1 = (k1 )u = 1 k t . As a result, the confinement effectiveness k cc of Eq. (6), can be
considered to be the lower bound value of k cc at all levels of confinement and k cc of Eq. (1), can
when
be considered to be its upper bound value.
2.1 Nonlinear Failure Envelopes of FRP-confined concrete
For a Mohr-Coulomb type maximum strength failure envelope, such as that introduced in Eqs.
(2)-(12) to properly model the behavior of concrete in a triaxial compression state of stress, it must
also include the cases of uniaxial tension, tension-compression and tension-tension stress states
and shall satisfy the following conditions: (1) It should pass through the point of uniaxial tension, i.e.
when
σ 3 = − f to , σ 1 = 0 ; (2) It should pass through the point of uniaxial compression, i.e. when
σ 3 = 0 , σ 1 = f co ; (3) It should fit the failure envelope of concrete in a triaxial compression state
of stress, and (4) It should approach the hydrostatic state when the confinement is very large (i.e.
as σ 2 = σ 3 → ∞ , σ 1 → σ 2 = σ 3 ), where σ 2 is an intermediate principal compressive stress.
The following Mohr-Coulomb failure envelope developed herein for confined concrete satisfies
these conditions. This model considers that the inherent curvature of the failure envelopes of
confined concrete, as shown in Fig. 1(b), can be attributed to the remolding of the internal structure
of the concrete core as damage in the concrete progresses which is attributed to the degradation of
the angle of internal friction φ c [9] of the confined concrete core, as shown in Fig. 3. As the
4
FRPRCS-8
University of Patras, Patras, Greece, July 16-18, 2007
internal structure of the concrete core degrades to its constituent materials (from internal crack
growth due to loss of adhesion provided by the cement paste), and is found to depend on the state
of stress ( σ n , σ 1 and σ 3 ) in the confined concrete core as follows:
Fig. 3 Plot of friction angle versus the normal stress on the failure shear plane of
the confined concrete core.
⎛ (k1 )be − 1 ⎞
⎟ ; ψ dil = φ o − φbe
⎟
kn
⎝ (k1 )be + 1 ⎠
1+
(k n )m
φ c = φ be + ψ dil ; φbe = sin −1 ⎜⎜
(k1 )be = neff
(13)
φ + φbe
1 + sin φ b
+ (1 − neff )(k1 )b ; (k1 )b =
(14)
; φm = o
1 − sin φb
2
φ
k
φ o = φu + σ (φ u − φbe )[1 + cos(2θ u )] ; φ u = 90° − 2 tan −1 k t ; θ u = 45° + u
2
kt
kσ =
cu
(σ n )m
=
k t (φ t − φ u )
2k t (φ t − φ be ) + (φ u − φ be )[1 + cos(2θ u )]
;
(k n )m =
where the basic angle of internal friction of the concrete
φb
(σ n )m
f co
=
(15)
(φu − φbe )[1 + cos(2θ u )] (16)
2(φ o − φ u )
is a material constant of the concrete
core that depends on the aggregate type (crushed stone or gravel), aggregate gradation and sandaggregate ratio of the concrete mix; for normal-weight normal strength concrete an average value
of φ b ≅ 36° is used herein, which was determined from analysis of dry triaxial compression
concrete cylinder tests [10]. In addition,
ψ dil =
angle of dilatancy of the concrete which depends
on the state of stress of the concrete core, and is governed by Eq. (13). In reference to Fig. 3,
φ o = initial angle of internal friction of the concrete core corresponding to the initial normal stress
(σ n )o = 0 ,
on the shear failure plane at the instant when
inclination of the failure shear plane corresponding to
coefficient;
(14);
(σ n )m =
(k n ) m
(σ n )o ;
σ n = 0 ; θo =
initial angle of
kσ = median angle normal stress
median-angle normal stress corresponding to the median angle
φm
of Eq.
= normalized median-angle normal stress; k n normalized normal stress in the
concrete core. In addition, as shown in Figs. 1 and 3, the apparent uniaxial cohesion cu , and the
uniaxial normal stress
(σ n )u
and the uniaxial angle of internal friction
5
φ u correspond to the
uniaxial compression case for which k 3e = σ 3e = 0 and σ 1 = f co . Excess pore water effects
[10,11] are accounted for in the angle of friction model of Eqs. (13)-(16) by utilizing the effective
basic confinement effectiveness coefficient (k1 )be of Eq. (14), the corresponding effective basic
angle of internal friction
φbe
of Eq. (13), and the effective porosity, neff , of the granular concrete
core with 0 < neff ≤ 1.0 . An average value of
an average value of
neff = 2.5% is used herein for FCC sections and
neff = 55.5% is recommended for CFFT sections.
2.2 Elastic and Plastic confinement effectiveness of FRP-confined concrete
Typical plots of the confinement effectiveness k c , versus the effective minor principal stress
k 3e , are shown in Fig. 4. Using Hooke’s law prior to cracking of the concrete core (i.e. the
elastic region), the elastic confinement effectiveness k ce , as shown in Figs. 4(b) and 4(d), can be
ratio
found as follows:
f
E
Eco
1 − µ do
k ce = ce = −ω E ε d ; ω E = k1e K o = do ; k1e =
; Ko =
(1 − µ do ) f co
f co
f co
µ do
µ do = −
[
]
∂ε d
E
= co = β d (µ B )o ; Eco = Eci 1 + α Eν ci (µ B )o (β sh )2 ; α E =
∂ε c E do
(µ B )o
=−
∂ε B
=
∂ε c
ν ci
[
1 + α E (1 + ν ci )1 − ν ci (β sh )2
(17)
(E je )B
(18)
Eci
; Eci ≈ 3320 f co + 6900 (19)
]
where f ce = average “elastic” stress in the concrete core; ω E = effective elastic confinement
index, which represents the average initial “elastic” slope of the confinement effectiveness versus
the effective minor principal stress ratio curves of Figs. 4(b) and 4(d); k1e = elastic confinement
effectiveness coefficient, and K o = normalized initial stiffness of the FRP-confined concrete.
Also, E co and E do are the initial diagonal and axial elastic modulus of the FRP-confined concrete
section, respectively; µ do and
minor
εB
(µ B )o
are the initial dilation rates along the diagonal
εd ,
and
strain directions, respectively. In addition, ν ci = initial Poisson’s ratio of the unconfined
concrete core, where 0.15 ≤ ν ci ≤ 0.25 ; the value
ν ci = 0.18
is used herein;
αE =
modular ratio; E ci = initial tangent modulus of the unconfined concrete core.
As shown in Figs. 4(a) and 4(c), in the plastic compressive region ( i.e. when
ε co ≤ ε cp ≤ ε cu ,
and
when
ε dp
where
ε do ≤ ε dp ≤ ε du
FRP jacket
ε c = ε cp
where
), the plastic confinement
effectiveness, k cp , of the FRP-confined concrete section can be found by rewriting k cc and k εc of
Eq. (1) as follows:
k cp =
ϕ1 p =
f cp
f co
k1 p
k1
= 1 + k1 p k 3e ; kεp =
; ϕ2 p =
k2 p
(R p )i k1
ε cp
ε co
; ϕR =
= 1 + k 2 p k 3e ; R p =
ϕ1 p
ϕ2 p
=
6
(R p )i
Rp
k2 p
k1 p
; ϕµ =
⎛ k2 p
; Rp = ⎜
i ⎜k
⎝ 1p
( )
µd
µ do
(20)
⎞
⎟ = 3 1 + α u (21)
⎟
⎠i
(
)
FRPRCS-8
University of Patras, Patras, Greece, July 16-18, 2007
Fig. 4 Typical dilation behavior of moderate stiffness FRP-confined concrete sections (a) Plot of
normalized dilation and confinement coefficients and (b) Plot of confinement effectiveness-versus
minor principal compressive stress-ratio
where f cp = axial plastic compressive stress at a given plastic compressive strain
corresponding plastic diagonal expansive strain
ε dp ; k1 p =
ε cp ,
and
plastic confinement effectiveness
coefficient; k 2 p = plastic strain coefficient; R p = plastic strain ductility ratio; k εp = plastic strain
( )i of
effectiveness [13,14,15]. The ideal plastic strain ductility ratio of FRP-confined concrete R p
( )i = 6.0 for concrete subjected to a triaxial compression state
Eq. (21), is considered equal to R p
of stress [10], as occurs when the section becomes more circular as
For high aspect ratio sections for which
ratio,
α sh >> 2.0
and
α sh → 1.0 and α j → 0.50 .
α j → 0 , the ideal plastic strain ductility
(R p )i , approaches that of concrete in a biaxial compression state of stress for which
(R p )i = 3.0 [12].
( )
In Figs 4(a) and 4(c) it can be observed that the ideal plastic confinement effectiveness k cp
i
represents the tangent of the confinement effectiveness k c , versus the effective minor principal
stress ratio k 3e , curve of the FRP-confined section when
( )i
can be given found by using by setting k cc = k cp
yields:
7
ϕ R = 1.0 (i.e. when ϕ1 p = ϕ 2 p ); and
in Eq. (6) and using Eqs. (7) and (9) which
(k cp )i =
( f cp )i
(22)
= k t k1 − ω p ε d ; ω p = α uψ 3γ shω je ; ω je = K je k1
f co
where ω p = ideal plastic confinement index of the rectangular FRP jacket, ω je = effective
confinement index of the FRP jacket.
The plastic confinement index
ωp
of Eq. (22) of the
rectangular FRP jacket (i.e. the tangent plastic slope shown in Figs. 4(b) and 4(d)), indicates the
upper bound tangent slope of the plastic confinement effectiveness k cp versus the effective minor
principal stress ratio k 3e , curve of the FRP-confined concrete when ϕ R = 1.0 or (i.e. when
ϕ1 p = ϕ 2 p ). The softening and flattening of the ϕ1 p , ϕ 2 p and ϕ R curves in Figs. 4(a) and
4(c), can be attributed to the softening of the mechanical response of the FRP-confined concrete
that results from remolding of the internal structure of the passively confined concrete core due to
transverse dilation of the confined concrete at high axial deformations as damage builds up. As the
lateral stiffness of the confining FRP jacket increases (i.e. higher the kinematic restraint) the
confinement effectiveness k c and the normalized plastic confinement coefficient ϕ1 p curves shift
from softening behavior to essentially bilinear hardening behavior.
3 UNIAXIAL STRESS-STRAIN MODEL FOR FRP-CONFINED CONCRETE
The typical axial compressive stress-strain behavior of rectangular FCC and CFFT members
can be modeled using a Popovics [7] type fractional stress-strain model introduced by the authors
[6] with the use of the ideal plastic strain ductility ratio R p of Eq. (21) and the ideal plastic
i
confinement effectiveness k cp of Eq. (22). In addition, the diagonal dilation of the rectangular
i
FRP confined concrete section can be modeled using a damage based dilation model introduced
by the authors [6] which considers that the dilation behavior of the FRP confined section is
dependent on the stiffness of the FRP composite jacket, the geometry of the section, the
mechanical properties of the unconfined concrete and FRP jacket, and the extent of internal
damage as measured by the expansive diagonal strain, ε d , in the confined concrete core. In the
( )
( )
analytical expressions developed herein, the only coefficients that were experimentally obtained
and used in the uniaxial stress-strain model are the average value of neff = 55.5% recommended
for CFFT sections, the average value of the basic angle of friction
concrete core, and the diagonal plastic dilation rate,
φb = 36°
of the confined
µ dp , of the dilation model introduced by the
authors [6].
4 COMPARISON OF MODEL WITH EXPERIMENTAL RESULTS
The stress-strain curves obtained using the damage based model presented in Eqs. (2)-(22) in
addition to the dilation and stress-strain model introduced by the authors [6] are compared to the
compressive tests of square and rectangular FCC tests [16] in Fig. 5(a), and [2] in Figs. 6(a) and
6(b), and to circular FCC tests [16] in Fig. 5(b).
From analysis of Figs. 5 and 6, it can be observed that the proposed damage-based model can
accurately capture both the strain softening behavior of low FRP jacket stiffness rectangular and
circular FRP-confined concrete sections, and the strain hardening behavior of moderate to high
FRP jacket stiffness rectangular and circular FCC concrete sections, as well as rectangular and
circular CFFT sections (not included herin dure to space limitations). In addition, the model can
accurately capture the effects of an increase in the aspect ratio of the rectangular FRP-confined
concrete section.
8
FRPRCS-8
University of Patras, Patras, Greece, July 16-18, 2007
Fig. 5 Comparison of analytical model predictions versus tests of (a) square and rectangular and
(b) circular FCC sections [16].
Fig. 6 Comparison of analytical model predictions versus tests of (a) square and (b) rectangular
FCC sections [2].
5 CONCLUSION
A uniaxial stress-strain model using basic principles of mechanics was introduced for
describing the compressive behavior and dilation behavior of circular and rectangular FRPconfined (FCC) concrete sections confined by bonded FRP jackets with rounded corners, and nonbonded (CFFT) sections. The distinguishing features of the proposed model are: (a) the use of a
damage-based variable diagonal secant Poisson’s ratio and dilation rate model for FRP-confined
concrete, which a function of the stiffness of the confining FRP composite jacket, the geometry of
the FRP-confined concrete section, the mechanical properties of the unconfined concrete core and
FRP jacket, and the extent of internal damage as measured by the expansive diagonal strain in the
FRP-confined concrete; and (b) the introduction of a Mohr-Coulomb type maximum strength failure
envelope for confined concrete, in which the confinement effectiveness of the confined concrete
was found to depend on the uniaxial tensile and compressive properties of the concrete material,
and the internal stress-dependent angle of internal friction of the confined concrete core.
6 ACKNOWLEDGEMENT
The authors would like to acknowledge financial support from the National Science Foundation
under contract No. CMS 0099792. The opinions expressed in this article are those of the writers,
and do not necessarily reflect the opinions of the sponsoring organization.
9
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