DEVELOPED AND IMPLEMENTED BY:
Jin-Yu LU, Southeast University, Nanjing, China
Guan LIN (guanlin@polyu.edu.hk), Hong Kong Polytechnic University, Hong Kong, China.
This command is used to construct a uniaxial hysteretic stress-strain model for fiber-reinforced polymer (FRP)confined concrete. The envelope compressive stress-strain response is described by a parabolic first portion and a
linear second portion with smooth connection between them (Figure 1). The hysteretic rules of compression are
based on Lam and Teng’s (2009) model. The cyclic linear tension model of Yassin (1994) for unconfined concrete (as
adopted in Concrete02) is used with slight modifications to describe the tensile behavior of FRP-confined concrete
(Teng et al. 2015).
Envelope in compression
Cyclic in compression (Lam and Teng 2009)
Envelope in tension (Yassin 1994)
Cyclic in tension (Yassin 1994)
Unconfined concrete
Stress − c
( − cu , − f cu )
− t
− fc0
Reloading curve
Unloading curve
Ec
Ets
− ft
− c 0
Eun ,0
Strain − c
Figure 1 Hysteretic Stress-Strain Relation
uniaxialMaterial FRPConfinedConcrete02 matTag $fc0 $Ec $ec0 <-JacketC $tfrp $Efrp $erup $R> <-Ultimate $fcu
$ecu> $ft $Ets $Unit
$matTag
integer tag identifying material
$fc0
compressive strength of unconfined concrete
$Ec
elastic modulus of unconfined concrete (= 4730√$𝑓𝑐0(𝑀𝑃𝑎))
$ec0
axial strain corresponding to unconfined concrete strength (≈ 0.002)
<-JacketC $tfrp $Efrp $erup $R>
input parameters of the FRP jacket in a circular section
$tfrp
thickness of an FRP jacket
$Efrp
tensile elastic modulus of an FRP jacket
$erup
hoop rupture strain of an FRP jacket
$R
radius of circular column section
<-Ultimate $fcu $ecu> input ultimate stress/strain directly
$fcu
ultimate stress of FRP-confined concrete ($fcu ≥ $fc0)
$ecu
ultimate strain of FRP-confined concrete
$ft
$Ets
$Unit
tensile strength of unconfined concrete (= 0.632√$𝑓𝑐0(𝑀𝑃𝑎))
stiffness of tensile softening (≈ 0.05 Ec)
unit indicator, Unit = 1 for SI Metric Units; Unit = 0 for US Customary Units
NOTES:
•
•
Compressive concrete parameters should be input as negative values.
The users are required to input either the FRP jacket properties in an FRP-confined circular column (<-JacketC>)
or directly input the ultimate point (εcu, fcu) (<-Ultimate>). If <-JacketC> is used, the ultimate stress and strain
are automatically calculated based on Teng et al.’s (2009) model which is a refined version of Lam and Teng’s
(2003) stress-strain model for FRP-confined concrete in circular columns. If <-Ultimate> is used, the ultimate
stress and strain can be calculated by the users in advance based on other stress-strain models of FRP-confined
concrete and thus can be used for other cross section shapes (e.g., square, rectangular, or elliptical). If none of
them is specified, a stress-strain curve (parabola + horizontal linear curve) for unconfined concrete will be
defined (Figure 1). Both <-JacketC> and <-Ultimate> adopt the envelope compressive stress-strain curve with
a parabolic first portion and a linear second portion.
•
Unit indicator: $Unit = 1 for SI Metric Units (e.g., N, mm, MPa); $Unit = 0 for US Customary Units (e.g., kip, in,
sec, ksi).
Calibration:
The implemented new material has been calibrated using a simple-supported Force-Based Beam-Column element
subjected to axial load only (http://opensees.berkeley.edu/wiki/index.php/Calibration_of_Maxwell_Material). The
output stress-strain responses were compared with the desired curves defined by the input parameters.
EXAMPLES:
Example 1: Pin-ended FRP-confined reinforced concrete (RC) columns
e
P
e
FRPConfinedConcrete02
l
Δ
Fiber section
ReinforcingSteel
e
(a) Column under eccentric loading
P e
(b) Idealized column model
Figure 2 Simulation of pin-ended FRP-confined RC column
The first example is a pin-ended FRP-confined circular RC column subjected to eccentric compression (load
eccentricity = 20 mm) at both ends tested by Bisby and Ranger (2010) (Figure 2). Due to the symmetry in geometry
and loading, only half of the column needs to be modelled. In this case, three forceBeamColumn elements each
with 5 integration points were used for the half column. The FRPConfinedConcrete02 model was used to describe
the stress-strain behavior of FRP-confined concrete. Either <-JacketC> or <-Ultimate> can be used. If the former is
used, the properties of the FRP jacket need to be input; if the latter is used, the ultimate stress and strain need to
be calculated by the users and input directly. The eccentric loading is applied with a combined axial load and bending
moment at each end node. An increasing vertical displacement is applied to the top node of the column model. The
analysis terminated until the ultimate axial strain of FRP-confined concrete was reached by the extreme
compression concrete fiber at the mid-height (equivalent to FRP rupture). SI Metric Unit (e.g., N, mm, MPa) is used
in the script of this example ($Unit = 1).
Figure 3 shows the comparison of axial load-lateral displacement curve between the test results and the theoretical
results. Figure 4 shows the variation of column slenderness ratio (l/D) on the axial load-lateral displacement
response of the column. Please refer to Lin (2016) for more details about the modeling.
700
Axial load P (kN)
600
500
400
300
200
Experimental results (C20)
100
Theoretical results
0
0
2
4
6
8
Mid-height lateral displacement Δ (mm)
10
Figure 3 Experimental results vs theoretical results
700
Axial load P (kN)
600
l/D = 2.0
500
l/D = 4.0
l/D = 6.0
400
300
200
e = 20.0
100
0
0
2.5
5
7.5
10
Mid-height lateral displacement Δ (mm)
12.5
Figure 4 Parametric study (effect of column slenderness ratio)
Example 2: Cantilever column subjected to constant axial compression and cyclic lateral loading
Axial load

Lateral load P
Beam-Column
Element
(2)
Unconfined region
FRPConfinedConcrete02
L
Fiber section
Fixed-end (1)
Rotation Element
ReinforcingSteel
FRP-confined region
(a) Single-curvature bending
(b) Fibre discretization of FRPconfined section
Figure 5 Simulation of column under cyclic latera loading
The second example is a cantilever FRP-confined circular RC column subjected to constant axial compression and
cyclic lateral loading (Column C5 tested by Saadatmanesh et al. 1997). The US Customary Units (e.g., kip, in, sec, ksi)
were used in this example. The twenty-five (25)-in.-height region (potential plastic hinge region) above the footing
of the column was wrapped with an FRP jacket; the remaining portion of the column with a height of 71 in. was
conventional RC section without FRP jacketing. The column was modelled using two forceBeamColumn elements to
cater for the variation of section characteristic along the column height. A zero length section element at the
column-footing interface was used to simulate fixed-end rotations due to the strain penetration of longitudinal steel
bars (Figure 5) (Lin et al. 2012). The bond-slip model of Zhao and Sritharan (2007) (Bond_SP01) was used to depict
the bar stress-slip response. In addition, another zero length section element was used at the column-footing
interface to consider the possible rotations of the footing (Teng et al. 2015). The rotation stiffness of the zero length
section element was adjusted to achieve close matching between the test response and the predicted response
during the initial stage of loading. This zero length section element was found to have little effect on the ultimate
displacement of the column (Teng et al. 2015). Moreover, the inclination of axial load in the column test needs to
be accounted for when comparing predicted results with test results (Teng et al. 2015). Figure 6 shows the
comparison of lateral load-lateral displacement curve between the test results and the theoretical results.
Lateral load P (kips)
20
10
0
-10
Experimental results (C5)
Theoretical results
-20
-4.5
-2.25
0
2.25
Tip lateral displacment Δ (in.)
4.5
Figure 6 Experimental results vs theoretical results
REFERENCES:
Bisby, L. and Ranger, M. (2010). “Axial-flexural interaction in circular FRP-confined reinforced concrete columns”,
Construction and Building Materials, Vol. 24, No. 9, pp. 1672-1681.
Lam, L. and Teng, J.G. (2003). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building
Materials, Vol. 17, No. 6, pp. 471-489.
Lam, L. and Teng, J.G. (2009). “Stress-strain model for FRP-confined concrete under cyclic axial compression”, Engineering
Structures, Vol. 31, No. 2, pp. 308-321.
Lin, G. (2016). Seismic Performance of FRP-confined RC Columns: Stress-Strain Models and Numerical Simulation, Ph.D.
thesis, Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong,
China.
Lin, G. and Teng, J.G. (2015). “Numerical simulation of cyclic/seismic lateral response of square RC columns confined with
fibre-reinforced polymer jackets”, Proceedings, Second International Conference on Performance-based and Lifecycle Structural Engineering (PLSE 2015), pp. 481-489 (http://plse2015.org/cms/USB/pdf/full-paper_7408.pdf).
Lin, G., Teng, J.G. and Lam, L. (2012). “Numerical simulation of FRP-jacketed RC columns under cyclic loading: modeling of
the strain penetration effect”, First International Conference on Performance-based and Life-cycle Structural
Engineering (PLSE2012), December 5-7, Hong Kong, China.
Saadatmanesh, H., Ehsani, M. and Jin, L. (1997). “Seismic retrofitting of rectangular bridge columns with composite straps”,
Earthquake Spectra, Vol. 13, No. 2, pp. 281-304.
Teng, J.G., Lam, L., Lin, G., Lu, J.Y. and Xiao, Q.G. (2015). “Numerical Simulation of FRP-Jacketed RC Columns Subjected to
Cyclic and Seismic Loading”, Journal of Composites for Construction, ASCE, Vol. 20, No. 1, pp. 04015021.
Yassin, M.H.M. (1994). Nonlinear Analysis of Prestressed Concrete Structures under Monotonic and Cyclic Loads, Ph.D.
thesis, University of California at Berkeley, California, USA.
Zhao, J. and Sritharan, S. (2007). “Modeling of strain penetration effects in fiber-based analysis of reinforced concrete
structuresconcrete structures”, ACI Structural Journal, Vol. 104, No. 2, pp. 133-141.