Numerical Investigation on Load-carrying Capacity of High-strength Concrete-encased Steel Angle Columns
International Journal of Concrete Structures and Materials
Numerical Investigation on Load-carrying Capacity of High-strength Concrete-encased Steel Angle Columns
To investigate the load-carrying capacity of high-strength concrete-encased steel angle (CES-A) columns, in which corner steel angles are encased in concrete and transmit column loads directly, a numerical study was performed by using a proposed analysis model. The proposed model considered the strain compatibility, confinement effect, local buckling, and premature cover-spalling, and was verified against previous experimental study results. To investigate the effect of design parameters, a parametric study was conducted, and based on the parametric study results, a simple approach was also discussed to predict the residual strength (2nd peak load) after spalling of concrete cover at corners (1st peak load). The numerical investigations showed that when steel contribution and confinement efficiency are high, CES-A columns exhibit relatively large load-carrying capacity even after cover-spalling, due to the maintained strength of confined concrete and yielding of steel angles, and the proposed simple approach gave a good prediction for the residual strength.
composite column; concrete-encased steel angle; high-strength steel; confinement; local-buckling; cover-spalling; residual strength
Steel–concrete composite columns such as
concrete-encased steel (CES) and concrete-filled steel tube (CFT)
columns have large load-carrying capacity and high local
stability due to composite action, and high-strength materials
improve structural safety and space efficiency. Thus, the use
of high-strength composite columns is growing in the
construction of high-rise and long-span structures.
When high-strength steel is used for conventional CES
columns (consisting of a wide-flange steel core and concrete
encasement), early crushing of concrete encasement needs to
be considered, because the steel core may not develop its full
plastic strength at the concrete failure, particularly under
(Kim et al. 2012, 2014)
. On the other hand,
CFT columns using high-strength steel show excellent
structural performance, because the steel tube provides good
lateral confinement to concrete core and the concrete core
restrains local buckling of the steel tube
(Kim et al. 2014)
However, in terms of fire proofing, local instability,
diaphragm connections, and concrete compaction, CES
columns still have advantages over CFT columns. Thus, further
studies are necessary for high-strength CES columns.
To improve the load-carrying capacity of CES columns,
concrete-encased steel angle (CES-A) sections can be used
(Kim et al. 2014, 2017; Eom et al. 2014; Hwang et al.
. In the case of CES-A sections (Fig. 1), the
contributions of steel to flexural strength and flexural
stiffness are maximized: the strain and moment-arm of steel are
significantly increased by placing steel angles at four
corners, and the corner steel angles connected by transverse
reinforcement provide good confinement to concrete core
(Kim et al. 2014, 2017)
. Because of these advantages, along
with constructability improvement by prefabrication, CES-A
sections have recently become popular in Korea.
Structural steel angles for columns have been widely
studied and used in various ways: (1) to externally
strengthen existing RC columns with batten plates (i.e., steel
jacketing for reconstruction or seismic retrofitting)
2001; Zheng and Ji 2008a, b; Montuori and Piluso 2009;
Nagaprasad et al. 2009; Calderon et al. 2009; Badalamenti
et al. 2010; Garzon-Roca et al. 2011a, b, 2012; Campione
2012a, b; Khalifa and Al-Tersawy 2014; Tarabia and
Albakry 2014; Cavaleri et al. 2016)
; (2) to replace
wideflange members with the built-up members connected by
(Hashemi and Jafari 2004)
; (3) to reduce laborious
fieldwork of composite columns by prefabrication
et al. 2011; Eom et al. 2014; Hwang et al. 2015, 2016)
or (4) to improve structural capacity and cost efficiency of
composite columns by maximizing the contribution of
(Kim et al. 2014, 2017)
. According to the
primary purpose, steel angles could be either encased in
concrete or exposed, and could transmit column loads or not
(providing confinement only). Especially for the steel
jacketing, extensive studies are available. However, the
discussions in the existing studies were limited to (1) low-strength
materials (because the steel jacketing is to strengthen
deficient columns) and (2) confinement of non-buckled steel
angles (because the externally attached steel angles are
generally not subjected to high compression), and (3) the
effect of concrete cover was not involved (because the
externally attached steel angles are exposed).
In the present study, to investigate the load-carrying
capacity of high-strength CES-A columns, in which corner
steel angles are encased in concrete and transmit column
loads directly (i.e., composite columns), a numerical study
was performed using a proposed analysis model. The
proposed model considered the strain compatibility,
confinement effect, local buckling, and premature cover-spalling.
For verification, the predictions by the proposed model were
compared with previous experimental study results
et al. 2014, 2017; Eom et al. 2014; Hwang et al. 2016)
to supplement the test results, a numerical parametric study
was conducted for various design parameters. To predict
residual strength after spalling of concrete cover at corners, a
simple approach was also discussed.
2. Previous Experimental Studies
Kim et al. (2014, 2017) performed eccentric axial load
tests for four CES-A columns with various transverse
reinforcements (lattices, battens, links, and rectangular spirals)
using Grade 800 MPa steel and 100 MPa concrete (Table 1;
Figs. 2, 3). The test results showed that even after spalling of
concrete cover at corners (1st peak) the axial load
continuously increased up to the 2nd peak without significant
degradation in stiffness, the maximum load and effective
flexural stiffness of the CES-A columns were 1.53 and 2.07
times those of a conventional CES column using a
wideflange steel core of the same area due to the high
contribution of steel angles, and the 2nd peak load was strongly
affected by confinement of transverse reinforcement and
Transverse reinforcement (see Fig. 3)a
C1–C5: Concentric axial load tests
F1–F7: Flexural tests
b = 260 mm, bc = 230 mm, Lk = 2620 mm for E’s
b = 500 mm, bc = 400 mm, Lk = 1500 mm for C1–C3 and F1–F4
b = 400 mm, bc = 300 mm, Lk = 1500 mm for C4–C5 and F5–F7
fc0;u = 103.6 MPa, eco;u = 0.0027, ecu;u = 0.003 for E1
fc0;u = 96.6 MPa, eco;u = 0.003, ecu;u = 0.003 for E2 and E3
fc0;u = 98.7 MPa, eco;u = 0.0032, ecu;u = 0.0032 for E4
fc0;u = 23.5 MPa, eco;u = 0.002, ecu;u = 0.003 for C’s and F’s
fys = 812 MPa, fus = 868 MPa, bs = 60 mm, ts = 15 mm for E1
fys = 759 MPa, fus = 884 MPa, bs = 60 mm, ts = 15 mm
for E2 – E4
fys = 444 MPa, fus = 689 MPa, bs = 90 mm,
ts = 7 mm for C’s and F’s
Lattices of /7 Bars (fyt = 531 MPa, st = 50 mm) for E1
Links of D13 Bars (fyt = 481 MPa, st = 100 mm) for E2
Battens of 6
60 mm Plates (fyt = 418 MPa, st = 210 mm) for E3
Spirals of D13 Bars (fyt = 567 MPa, st = 100 mm) for E4
Links of D10 Bars (fyt = 522 MPa) for C’s and F’s
st = 100 mm for C2, C4, F2, and F5
st = 200 mm for C1, C3, C5, F1, F3, and F6
st = 300 mm for F4 and F7
8-D13 (fyl = 513 MPa, ful = 634 MPa) for E1
4-D19 (fyl = 523 MPa, ful = 650 MPa) for C1 and F1
aElastic modulus = 3320 fc0 þ 6900 (MPa) for concrete, 205 GPa for steel plates including steel angles and battens, and 200 GPa for
reinforcing steel bars including lattices, links, and spirals.
steel angles and local buckling of steel angles.
Eom et al.
Hwang et al. (2016)
performed concentric axial
load tests and flexural tests for twelve CES-A members
(yield strength of steel fys = 444 MPa and compressive
strength of concrete fc0 = 23.5 MPa). The test parameters
included the sectional ratio of steel angles and vertical
spacing of links to investigate the confinement effect and the
flexural, shear, and bond resistances (Table 1; Figs. 2, 3).
Additionally, to evaluate seismic performance, Hwang et al.
(2015, 2016) conducted cyclic load tests for CES-A columns
and beam-column joints. The test results showed that, in the
case of the concentrically loaded specimens, the uniaxial
strength was greater than the design strength Po ¼
0:85fc0Ac þ fysAs þ fylAl (Ac, As, Al = area of concrete, steel,
or longitudinal bars, and fyl = yield strength of longitudinal
bars) due to the confinement effect. In the case of the
flexurally loaded specimens, crushing of concrete cover
(corresponding to the ultimate compressive strain of ecu = 0.003)
did not control the maximum deformation due to the
confinement effect, and ductile behavior was maintained until
steel angles reached their fracture strain (ecs = 0.015).
Also, the previous experimental study results
(Kim et al.
2014, 2017; Eom et al. 2014; Hwang et al. 2015, 2016)
showed that the strain compatibility method of ACI 318-14
(2014) with ecu = 0.003 underestimated the load-carrying
capacity of the CES-A columns by neglecting the
confinement effect of steel angles and transverse reinforcement on
concrete core. On the other hand, the load-carrying capacity
was less than the prediction by the plastic stress distribution
method of Eurocode 4 (2005) due to failure (spalling or
crushing) of concrete cover earlier than yielding of steel
angles. As such, since the load-carrying capacity of CES-A
columns is strongly affected by the behavior of concrete
cover and steel angles, further studies are required for those
3. Nonlinear Numerical Analysis
To investigate the load-carrying capacity of CES-A
columns with various design parameters, nonlinear numerical
analysis was performed. To this end, the fiber model analysis
method was used
(El-Tawil and Deierlein 1996; Kim et al.
2012, 2014, 2017)
considering the strain compatibility,
confinement effect of steel angles and transverse
reinforcement, local buckling of steel angles and longitudinal bars,
premature spalling of concrete cover at corners, effect of
local buckling of steel angles on confinement, and
secondorder effect (Fig. 4). In the analysis model, a composite
section was divided into four regions to evaluate the
contribution of each structural component: unconfined concrete
cover, confined concrete core, steel angles, and longitudinal
bars. The detailed modeling equations and analysis
procedure are given in the following subsections.
3.1 Concrete and Confinement
Since the distribution of confining pressure in CES-A
sections is similar to that of rectangular tied RC sections,
existing concrete confinement models can be used, regarding
steel angles and transverse reinforcement as longitudinal
bars and transverse bars, respectively
(Montuori and Piluso
2009; Nagaprasad et al. 2009)
. In the present study, the
compressive stress–strain relationship of concrete was
characterized by the confinement model of Saatcioglu and
(Saatcioglu and Razvi 1992; Razvi and Saatcioglu
(Fig. 4a, b), which is applicable to a wide range of
materials (concrete of fc0 = 30 – 130 MPa and transverse
reinforcement of up to fyt = 1400 MPa), uses the actual
tensile stress of transverse reinforcement at the peak stress of
confined concrete (instead of assuming yielding of
transverse reinforcement), and was verified against extensive test
results of columns under concentric and eccentric loadings.
In the model, the strength and ductility of confined concrete
are defined as a function of the equivalent uniform lateral
confining pressure rle, and the tensile stress of concrete is
ignored (Eq. (1)).
8 h i
fc ¼ : fc0 1
< fc0hr 1rðþecð=ece=coeÞcoÞr
0:15 eec8c5 eecoco
if ec [ eco
where fc and ec = concrete stress and corresponding strain;
fc0 and eco = peak stress and corresponding strain (or peak
strain); ec85 = strain corresponding to 0.85 fc0 after peak
stress; pffiffiffir ¼ Ec= Ec Ec;sec = modular ratio;
Ec ¼ 3320 fc0 þ 6900 = elastic modulus; and
Ec;sec ¼ fc0=eco = secant modulus at peak stress (Ec;sec Ec).
Calculations of these material properties for unconfined
concrete and confined concrete are available in the literature
(Saatcioglu and Razvi 1992; Razvi and Saatcioglu 1999)
summarized in Appendix.
To take into account the distinctive local failure
mechanisms of CES-A columns in the analysis, the concrete model
was modified as follows.
Concrete cover is a protection of steel and
reinforcement against corrosion and fire
, and premature spalling of concrete
cover may occur due to shrinkage of concrete and
weakness planes between concrete cover and concrete
core, which are created by longitudinal and transverse
(Collins et al. 1993; Cusson and Paultre
. This phenomenon is more obvious when higher
strength concrete and denser reinforcement are used
(Collins et al. 1993; Cusson and Paultre 1994)
Especially in CES-A columns, the premature
coverspalling is more pronounced at corners due to the
smooth surface of steel angles
(Kim et al. 2014, 2017)
Thus, for the stress fc;u of concrete cover at corners, the
proposed model assumed Eq. (2) (case 1 in Fig. 5). In
the absence of experimental data, the ultimate strain of
ecu;u = 0.003 is recommended based on test results
(Kim et al. 2014, 2017)
fc;u ¼ 0 if ec [ ec;u for concrete cover at corners
(2) Steel angles provide good confinement to concrete core
(Calderon et al. 2009; Montuori and Piluso 2009;
Nagaprasad et al. 2009; Badalamenti et al. 2010; Kim
et al. 2014, 2017; Eom et al. 2014; Hwang et al.
, but the effect of local buckling on the
confinement should be also considered. Thus, unlike
the existing models
(Calderon et al. 2009; Montuori
and Piluso 2009; Nagaprasad et al. 2009; Badalamenti
et al. 2010)
, in which the full leg bs of steel angles was
assumed to exert the confining pressure in whole
analysis steps, the proposed model assumed that only
the effective leg bs;eff exerts the confining pressure after
local buckling of steel angles (Fig. 6). The effective leg
also varies in vertical, but concrete failure occurs at the
weakest point (i.e., at the mid-height within a buckling
length). Thus, in the calculation of the geometrical
effectiveness coefficient k2 (see Appendix), Eq. (3) was
used for the ineffective width wi (bc = dimension of
confined concrete) to consider the reduction in
confining pressure by local buckling.
wi ¼ bc
It is noted that when battens are used for transverse
reinforcement, the center-to-center spacing st can be substituted
by an effective spacing in the calculation of k2
and Piluso 2009; Nagaprasad et al. 2009)
. Thus, the clear
spacing sc ¼ st ht (ht = height of a batten) was used as
the effective spacing. Calculation of the effective leg bs;eff
considering local buckling of steel angles under non-uniform
compression is given in the next subsection. Figure 7 shows
an example for the confining pressure rle under pure
compression. As shown in the figure, the confining pressure is
gradually decreased after local buckling of steel angles. The
reduction of the confining pressure due to the local buckling
has an effect on post-peak behavior (or ductility) of CES-A
3.2 Steel Angles and Local Buckling
The stress–strain relationship of high-strength steel angles
was characterized by a rounded curve (Fig. 4c and Eq. (4))
(Ramberg and Osgood 1943; Rasmussen 2003)
. For mild
steel angles, a typical trilinear curve (with elastic, plastic,
and strain-hardening ranges) can be used (Zubydan and
< Efss þ 0:002 ffyss n if fs fys
fs fys m for high-strength steel
es ¼ : fsE0:f2ys þ eus fus fys þeys if fys\fs fus
where fs and es = steel stress and corresponding strain; fys
and fus = yield and ultimate stresses; eys and eus = yield and
ultimate strains; and Es = elastic modulus. For high-strength
steel, 0.2% proof stress can be used as the yield stress and
E0:2 ¼ Es= 1 þ 0:002n= fys=Es = tangent modulus at the
0.2% proof stress; eys ¼ fys=Es þ 0:002;
n ¼ 1 0:2 þ 185fys=Es fus=fys =0:0375 þ 5; and
m ¼ 1 þ 3:5fys=fus.
Generally, in the design of conventional CES columns,
local buckling of the steel core is not considered, because
concrete encasement prevents the local buckling. However,
in the case of CES-A columns, the structural behavior after
spalling of concrete cover at corners is strongly affected by
local buckling of steel angles. Particularly, since composite
columns are likely to be subjected to eccentric loads and
relatively thick steel sections are preferred to support
construction loads before concrete develops its design strength,
much concern should be given to inelastic local buckling
under non-uniform compression. The critical buckling stress
fbs in elastic and inelastic ranges can be determined by the
well-known formula of Eq. (5)
, and the
postlocal buckling behavior can be described by the effective
(von Karman et al. 1932; Winter 1947;
Shanmugam et al. 1989; Usami 1993; Bambach and
Rasmussen 2004a, b; Bambach and Rasmussen 2004; Liang
et al. 2006; Mazzolani et al. 2011)
. In the effective width
approach, the stress redistribution within a buckled steel
plate is simplified by assuming that only a certain width of
the plate remains effective.
fbs ¼ g
where g = plasticity reduction factor; kb = local buckling
coefficient; m = Poisson’s ratio; and bs and ts = width and
thickness of steel, respectively.
The plasticity reduction factor was calculated by Eq. (6)
Mazzolani et al. (2011)
experimentally verified that the equation gives a good
prediction for the critical load of steel angles) and the Poisson’s
ratio in elastic and inelastic ranges was calculated by Eq. (7)
(Gerard and Becker 1957; Mazzolani et al. 2011)
Fig. 7 Reduction of confining pressure after local buckling of
Es;sec 1 2
Es 3 þ 3
v ¼ vp
4 þ 4 Es;sec
where Es, Es;sec, and Es;tan = elastic, secant (¼ fs=es), and
tangent (¼ dfs=des) moduli; and me and mp = elastic (= 0.3)
and fully plastic (= 0.5) values of Poisson’s ratio,
The legs of steel angles can be modeled as a plate simply
supported along three edges with one longitudinal edge free
(Timoshenko and Gere 1985)
, and Bambach and Rasmussen
aSS = simply supported edge, Free = free edge, f1, f2 = edge stresses of a plate element (f1 f2), and w ¼ f2=f1 = ratio of edge stresses
( jwj 1).
bTo take into account the effect of transverse reinforcement, the second term Ck ðbs=stÞ2 was newly introduced based on the results of finite
strip analysis and regression analysis. When battens are used for transverse reinforcement, the center-to-center spacing st can be substituted by
the clpearffiffiffiffisffipffiffiffiaffifficing sc ¼ st ht (ht = height of a batten).
ck ¼ fy=fbs, and a = 0.22 = imperfection sensitivity coefficient to consider initial imperfections.
(2004a, b) proposed two methods (elastic and plastic
effective width methods) for the plate element to consider stress
gradient and initial imperfections (geometric imperfection
and residual stress). Of the two, the elastic effective width
method (Table 2; Fig. 8) was implemented in the present
study: although the elastic effective width method
underestimates the ultimate flexural strength of the plate element
(Bambach and Rasmussen 2004a, b)
, it is consistent with
current design codes
(Bambach and Rasmussen 2004a, b)
and more suitable to calculate the geometrical effectiveness
coefficient k2 of confined concrete.
Bambach and Rasmussen (2004a, b) calculated the local
buckling coefficient by finite strip analysis with large
halfwavelengths (or for long plates). However, since the local
buckling coefficient is affected by boundary conditions and
plate geometry, the effect of transverse reinforcement needs to
(Timoshenko and Gere 1985)
. Thus, the local
buckling coefficient was modified using CUFSM
, which is an open source finite strip elastic
stability analysis program, and regression analysis. The modified
equations for the local buckling coefficient considering the
spacing of transverse reinforcement are given in Table 2 (the
newly introduced second term Ck ðbs=stÞ2 is the modification).
The critical buckling strain ebs of steel angles was assumed
to be greater than the peak strain eco;u of concrete cover (i.e.,
ebs eco;u) because concrete cover restrains local buckling of
(Chen and Lin 2006)
, and local bucking of steel
angles was assumed to incorporate spalling of concrete cover
at corners (i.e., fc;u ¼ 0 if es ebs) (case 2 in Fig. 5).
3.3 Longitudinal Bars and Local Buckling
For longitudinal bars of mild steel, a trilinear stress–strain
curve was used (Fig. 4d). In the figure, eyl, ehl, eul, fyl, fhl, ful are
yield, hardening, and ultimate strains, and their corresponding
stresses, respectively (assuming ehs ¼ 10eys and eus = 0.15 for
(Zubydan and ElSabbagh 2001)
). Local buckling of
longitudinal bars was assumed to initiate when the strain of
longitudinal bars reaches the peak strain of concrete cover (i.e.,
ebl ¼ eco;u)
(Gomes and Appleton 1997; Chen and Lin 2006)
and post-local buckling behavior was modeled by the plastic
mechanism approach (or yield-line analysis, Eq. (8))
et al. 1986; Gomes and Appleton 1997)
fl ¼ fbl4
3psbl X 5
where el and fl = strain and corresponding stress;
fbl = stress at the onset of local buckling (el ¼ ebl);
X ¼ 1
sbl = local buckling length (assumed to be the vertical
spacinqgffiffiffiffiffiffiffiffiffiffiffiffisffiffitffiffiffiffiffiffiffiffiffiffiffiffioffiffiffifffiffiffiffiffiffiffiffiffi transverse reinforcement);
eblÞ 2; and Dl = diameter of
longitu3.4 Bond Strength between Concrete and Steel
In the case of CES-A sections, frictional bond between
concrete and steel is insignificant due to the smooth surface
of steel angles, but transverse reinforcement provides two
mechanical bond mechanisms (concrete bearing and dowel
action). The authors experimentally investigated the bond
resistance of CES-A beams
(Eom et al. 2014)
, and it was
found that the bond resistance is mainly provided by
concrete bearing at a small flexural deformation and by dowel
action at a large flexural deformation, and bond-slip of
tension steel angles affects the structural behavior. Thus, to
prevent premature bond-slip, a denser spacing of transverse
reinforcement needs to be used. However, in the case of
CES-A columns, the effect of bond-slip is less pronounced
because large axial compression is expected to apply and
external forces are transferred to concrete and steel angles
directly. Thus, in the present study, the bond-slip was not
considered for simplicity.
3.5 Second-order Effect
The overall behavior of a column is affected by the
second-order effect. The second-order effect can be
approximately taken into account by assuming a cosine curve for the
deflection shape of the column (Fig. 4e)
. For the given eccentricity e0 and
compressive strain ec, the curvature jm at the mid-height section was
determined by iterations so that the force equilibrium of
Eq. (9) is satisfied.
Mm ¼ RðfiAiyiÞ ¼ Pðe0 þ DmÞ
where Mm = moment at the mid-height section; fi, Ai,
yi = stress, area, and moment-arm (from the centroid of the
composite section) of each fiber element; P ¼ RðfiAiÞ =
axial load; Dm ¼ jm=ðp=Lk Þ2 = lateral deflection at the
mid-height section; and Lk = buckling length.
For verification, the nonlinear numerical analysis results
by the proposed model were compared with the previous test
(Kim et al. 2014; 2017; Eom et al. 2014; Hwang et al.
. Table 1 and Figs. 2, 3 present the material and
geometric properties, section types, and transverse
reinforcement types of the previous test specimens.
Figure 9 shows the comparison. Although some
discrepancies were observed in the behavior, the predictions (thin
dashed lines) generally agreed well with the test results
(thick solid lines): the mean and standard deviation of
prediction-to-test ratios were 1.02 and 0.10 for the maximum
load (the larger of the 1st and 2nd peak loads) or 1.09 and
0.35 for the secant stiffness at the maximum load (Table 3).
The discrepancies in the behavior are mainly attributed to
local failures of the test specimens such as weld-fracture
between steel angles and lattices in E1 (Fig. 9a)
(Kim et al.
, premature spalling of concrete cover in C1 – C5
(Fig. 9e–i) (Hwang et al. 2016), premature tensile fracture of
steel angles in F3 (Fig. 9l)
(Eom et al. 2014)
, and premature
bond-slip of tension steel angles in F4, which is a beam
specimen (Fig. 9m)
(Eom et al. 2014)
. It is noteworthy that
the onset of local buckling of steel angles resulting from the
numerical analysis (marked by hollow triangles)
corresponded fairly well to the sudden drop or abrupt
coverspalling in the post-peak behavior of C1–C5 and F1–F7
(marked by solid triangles). Although the exact strain and
location of local buckling in the actual test specimens could
not be detected (because steel angles were encased in
concrete and the relatively thick steel angles did not buckle
enough to recognize), this agreement also confirms the
validity of the proposed model.
For more detailed investigation, the strength contributions
of unconfined (cover) concrete, confined (core) concrete,
steel angles, and longitudinal bars obtained from numerical
analysis are separately presented in Fig. 9 (thin dashed lines
with markers). In the case of the eccentrically loaded
specimens (E1–E4 in Fig. 9a–d), the axial load reached its
maximum even after cover-spalling (see the marker of u) due
to the maintained strength of confined concrete (see the
marker of c) and yielding of steel angles (see the marker of
s). On the other hand, in the case of the concentrically loaded
specimens (C1–C5 in Fig. 9e–i), cover-spalling determined
the maximum load, since it occurred around the entire
perimeter. In the case of the flexurally loaded specimens
(F1–F7 in Fig. 9j–p), the flexural strength was maintained
after cover-spalling due to the ductile behavior of confined
concrete and steel angles.
The large load-carrying capacity of CES columns after
failure of concrete cover (and local buckling of longitudinal
bars) was also reported by
Naito et al. (2011)
, and this
beneficial effect could be more pronounced in the case of
using high-strength and compact steel angles in CES-A
4. Parametric Study and Discussion
To investigate the effect of design parameters, a parametric
study was performed for a typical CES-A column shown in
Fig. 10 (square cross-section with equal-leg steel angles
connected by battens and having 50 mm-thick concrete
cover). The design parameters included the compressive
strength of concrete (fc0;u = 40, 60, and 80 MPa), yield
strength (fys = 315, 450, and 650 MPa, corresponding to
SM490 (ultimate strength fus = 490 MPa), SM570
(fus = 570 MPa), and HSA800 (fus = 800 MPa) in the
Korean Standards, respectively), area (or sectional ratio of
steel to gross section qs ¼ As=Ag = 3.2, 3.8, and 4.8%,
which correspond to bs ts = 150 10, 150 12, and 150
15 mm), and compactness (or width-to-thickness ratio of
the outstanding leg b0s=ts ¼ ðbs tsÞ=ts = 7.0, 11.5, and
16.5 under the same steel ratio of qs = 3.8%, which
correspond to bs ts = 120 15, 150 12, and 175
10 mm) of steel angles, yield strength (fyt = 315, 450, and
650 MPa), thickness (bt tt = 100 10, 100 12, and
100 15 mm), and spacing (st=bc = 0.3, 0.5, and 1.0) of
battens, eccentricity of axial load (e0=b = 0.1, 0.3, 0.5, and
1.0), column length (Lk = 4, 5, and 6 m), and size of
crosssection (bx by = 500 500, 600 600, and 700
700 mm). In the parametric study, the peak strain of
unconfined concrete was assumed to be eco;u = 0.0023,
0.0026, or 0.0028 for 40, 60, or 80 MPa concrete
Committee for Standardization 2008)
. The rounded stress–
strain relationship was used for 450 or 650 MPa steel,
whereas the trilinear relationship was used for 315 MPa
steel. The default (controlled) parameters are presented in
the bottom of Fig. 10.
4.1 Effects of Design Parameters
Figure 10 shows the axial load–strain (P ec)
relationships for the various design parameters. To quantify the
effect of each design parameter, the residual strength ratio a,
the ratio of the 2nd peak load (or residual strength) to the 1st
peak load (or spalling load), was introduced (presented in
Since the 2nd peak load is developed by the maintained
strength of confined concrete and yielding of steel angles,
generally a was increased as the steel contribution and
confinement efficiency increased. In more detail, as the
concrete strength fc0;u increased, a was decreased (a = 1.09
for fc0;u = 40 MPa, 1.02 for 60 MPa, and 0.98 for 80 MPa in
Fig. 10a), since the use of higher strength concrete resulted
in higher strength-loss by cover-spalling and less ductile
behavior. As the yield strength fys or sectional ratio qs of
steel angles increased, a was increased (a = 0.94 for
fys = 315 MPa, 0.96 for 450 MPa, and 1.02 for 650 MPa, or
a = 0.98 for qs = 3.2%, 1.02 for 3.8%, and 1.08 for 4.8%
in Fig. 10b). However, as the width-to-thickness ratio
increased, a was decreased (a = 1.06 for b0s=ts = 7.0, 1.02
for 11.5, and 1.01 for 16.5 in Fig. 10b), because the slender
section was more vulnerable to local instability. As the yield
strength fyt and thickness tt of battens increased, a was
slightly increased, but their effects were not significant
(Fig. 10c). On the other hand, the spacing st of battens had a
great effect (a = 1.15 for st=bc = 0.3 (clear spacing
sc ¼ st ht = 50 mm, and volumetric ratio of battens to the
confined concrete core qt ¼ RðbtttLtÞ= bc;xbc;yst = 6.25%:
Lt = length of battens), 1.02 for 0.5 (sc = 150 mm,
aT and A indicate the test and analysis results evaluated at the mid-height section.
qt = 3.75%), and 0.97 for 1.0 (sc = 400 mm, qt = 1.88%)
in Fig. 10c). It is because that the spacing st of battens is
related not only to confinement but also to local buckling of
steel angles. In comparison with the effect of battens
(Fig. 10c), the effect of steel angles (Fig. 10b) was also
highly influential for the load-carrying capacity of CES-A
columns. This parametric study result partly differs from the
result of an existing study: in the case of RC columns
strengthened by steel jacketing, thick and/or dense battens
are much more effective than large steel angles for
(Khalifa and Al-Tersawy 2014)
partly different result comes from the different purpose of
steel angles: unlikely the existing study for steel jacketing, in
which the primary purpose of steel angles is to provide
lateral confinement, the steel angles in CES-A columns are
used to transmit column loads directly as well as to provide
confinement. Thus, the steel angles in CES-A columns are
subjected to high compression, and their properties
associated with local buckling and confinement are also important
for load-carrying capacity. The eccentricity e0 of axial load
also had a significant effect, but a was not directly
proportional to e0 (a = 0.96 for e0=b = 0.1, 1.02 for 0.3, and 1.06
for 0.5, whereas a = 1.06 for a high eccentricity of
e0=b = 1.0 in Fig. 10d): under the higher eccentricities (or
lower compression), strength-loss by cover-spalling was less
pronounced, but the contribution of steel angles to axial load
was decreased due to the increased bending moment. As the
column length Lk increased, a was decreased (a = 1.02 for
Lk = 4 m, 1.00 for 5 m, and 0.98 for 6 m in Fig. 10e) due to
the increased slenderness and 2nd-order effect. On the other
hand, the effect of the increased slenderness by using a
smaller section was compensated by the increased steel
contribution and confinement efficiency in the smaller
section (a = 1.08 for b = 500 mm, 1.02 for 600 mm, and 0.99
for 700 mm in Fig. 10e).
4.2 Spalling Load and Residual Strength
The passive confining pressure is generated by the
laterally expanding concrete under compression due to the
Poisson effect and the restraining forces in steel angles and
(Saatcioglu and Razvi 1992; Razvi
and Saatcioglu 1999; Nagaprasad et al. 2009; Badalamenti
et al. 2010)
. Thus, the confinement effect is not fully
developed until a column is subjected to sufficient
compression and deformation, and the large compression and
deformation lead to cover-spalling (American Concrete
Institute 2014). As stated in the subsection of Concrete and
Confinement, cover-spalling is more pronounced at the
corners of CES-A columns. However, in well-confined
sections, strength-gain in confined concrete may compensate
or even exceed strength-loss in concrete cover
. Furthermore, as stated in the subsection of
Verifications, in the case of using high-strength and compact
steel angles, strength-gain after cover-spalling is more
pronounced. That is, the 2nd peak load can be even greater than
the 1st peak load depending on the steel contribution and
Figure 11 shows the numerical P M interaction curves
of the typical CES-A section (Fig. 10), which correspond to
the 1st peak load (thick dashed lines) and 2nd peak load
(thick solid lines). As expected, the 2nd peak load (or
residual strength) was affected by the design parameters, and
in some cases, the 2nd peak load was greater than the 1st
peak load: (1) as the steel contribution increased (in the cases
of using higher strength (Fig. 11c) and/or larger (Fig. 11d)
steel angles), the residual strength in the tension-controlled
zone (below the balanced failure point) was increased (by
comparing with Fig. 11a or the shaded area in each figure);
whereas (2) as the confinement efficiency increased (in the
cases of using more compact steel angles (Fig. 11e), higher
strength, thicker, and/or denser battens (Fig. 11f–h), the
residual strength in the compression-controlled zone (above
the balanced failure point) was increased. In the case of
using higher strength concrete (Fig. 11b), the interaction
curve expanded toward the compression-controlled zone, but
the residual strength was decreased due to the decreased
steel contribution. Especially in the practical range of axial
load (generally in actual design, P 0:1Agfc0;u according to
the definition of compression members and e0=b 0:1 to
account for accidental eccentricity
), the residual strength was obviously greater
than the 1st peak load.
The 2nd peak load or residual strength is a meaningful
factor in seismic design and progressive collapse analysis.
Thus, a rational approach is required to predict the residual
strength after cover-spalling. It is noted that the 1st peak load
(or spalling load) can be obtained by the strain compatibility
method of ACI 318-14
(Kim et al. 2014; 2017)
, in which the
linear strain distribution and ultimate compressive strain of
ecu = 0.003 for concrete are used neglecting the
confinement effect (American Concrete Institute 2014).
4.3 Simple Approach for Residual Strength
To construct the P M interaction curve corresponding to
the 2nd peak load (or residual strength) of CES-A sections, a
simple approach was proposed: for more exact
strengthcalculation and wider application, a strain-based method was
used. To accurately predict the strength of composite
sections with high-strength steel and high-level confinement,
the strain compatibility and confinement effect should be
(Kim et al. 2014, 2017)
. For the confinement
effect, the design equations of Eurocode 2
Committee for Standardization 2008)
were used with some
modifications: in Eurocode 2, parabola-rectangle stress–
strain relationships are provided for unconfined concrete and
confined concrete, and the peak stress fc0;c, peak strain eco;c,
and ultimate strain ecu;c of confined concrete are defined as a
function of the effective confining pressure rle (Eq. (10)).
fc0;u 1:125 þ 2:5 frc0;lue
if frc0;lue [ 0:05
< fc0;u 1 þ 5 frc0;lue
eco;c ¼ eco;u
ecu;c ¼ ecu;u þ 0:2 rle
where the peak and ultimate strains of unconfined concrete
are eco;u = 0.002 and ecu;u = 0.0035 for 12 fc0;u\
50 MPa, or eco;u ¼
where the subscript i indicates the i-th fiber in the
crosssection. The strain distribution or the strain ei of i-th fiber
will be discussed below.
The numerical investigation showed that the residual
strength is determined by (1) failure (local buckling) of steel
angles or (2) failure (crushing) of confined concrete,
whichever is earlier. The failure criteria can be defined as a
function of the strain of steel angles or confined concrete.
For a square section with equal-leg angles connected by
battens, the critical buckling strain ebs of steel angles and the
crushing strain ecu;c of confined concrete can be calculated as
The critical buckling strain ebs in the elastic range
(g = 1 from Eq. (6) because Es;sec ¼ Es;tan ¼ Es,
m = 0.3 from Eq. (7), and fbs ¼ Esebs in Eq. (5)) and
inelastic range (g ¼ 2=3 fys=ebs =Es from Eq. (6)
because Es;sec ¼ fys=ebs and Es;tan = 0, m = 0.5 from
Eq. (13): ebs ¼ ebs1 if ebs1
ebs1 [ eys.
Eq. (7), and fbs ¼ fys in Eq. (5)) can be rewritten as
eys, or ebs ¼ ebs2 if
ebs1 ¼ 12ð1
where the local buckling coefficient can be conservatively
taken as kb ¼ 0:43 þ ½bs=ðst htÞ 2 (for battens) from
(2) The effective confining pressure rle on confined
concrete can be rewritten as Eq. (14) from the
subsection of Concrete and Confinement and Appendix, and
then ecu;c can be obtained by using rle and Eq. (10c).
rle ¼ k2qtft
¼ st wiðst htÞ
Et 0:0025 þ 0:04 3
fc0;ust wiðst htÞ
where k2 1:0, ft fyt, At1 ¼ ht tt = area of a batten,
wi ¼ bc 2bs (before local buckling), and Et = elastic
modulus of battens.
When the extreme compression fiber of steel angles or
confined concrete (distance from the compression surface
yi ¼ dc) reaches its failure strain (ecf = the smaller of ebs
and ecu;c), the 2nd-peak load (residual strength) is developed.
Thus, with the assumption of linear strain distribution, the
strain distribution or the strain ei of i-th fiber can be
calculated as Eq. (15).
ei ¼ ecf
dcÞ ¼ ecf
dc ð i
where et = strain of the extreme tension fiber at yi ¼ dt.
The interaction curve for the residual strength can be
obtained by increasing et and summing up internal forces
(axial force P and bending moment M ) over the
cross-section. As shown in Fig. 11, the interaction curve for the
residual strength by the simple approach (thin solid lines)
agreed well with that by the numerical analysis (thick solid
lines) for all cases.
It is noted that, in the case of using the residual strength
for design purpose, the partial factors for materials are
recommended to be used: fc0u;d ¼ fc0u=1:5, fc0c;d ¼ fc0c=1:5, and
fys;d ¼ fys=1:1
(European Committee for Standardization
. Even though CES-A columns showed good
performance under cyclic loading
(Hwang et al. 2015, 2016;
Zheng and Ji 2008a, b)
, further studies on ductility and
postyield stiffness as well as residual strength are required for
seismic design and progressive collapse analysis.
To investigate the load-carrying capacity of high-strength
CES-A columns, in which corner steel angles are encased in
concrete and transmit column loads directly, a numerical
study was performed using a proposed analysis model. The
findings of the numerical study are summarized as follows.
(1) Considering the strain compatibility, confinement
effect of steel angles and transverse reinforcement,
and local buckling of steel angles and longitudinal bars,
nonlinear numerical analysis was performed. In the
analysis, the premature spalling of concrete cover at
corners and the effect of local buckling of steel angles
on confinement, which are the distinctive local failure
mechanisms of CES-A columns, were also taken into
(2) For verification, the numerical analysis results were
compared with the previous experimental study results.
The proposed model gave fairly good predictions for
the peak load, secant stiffness at the peak load, and
post-beak behavior. To investigate the effect of design
parameters (strength of concrete; strength, area, and
compactness of steel angles; strength, thickness, and
spacing of battens; eccentricity of axial load; and
slenderness by varying column length and sectional
size), a parametric study was also conducted.
(3) The numerical investigation showed that when the steel
contribution is high (by using higher strength, and/or
larger steel angles; or by using lower strength concrete)
and the confinement efficiency is high (by using more
compact steel angles; or by using higher strength,
thicker, and/or denser battens), CES-A columns exhibit
relatively large load-carrying capacity even after
spalling of concrete cover at corners due to the
maintained strength of confined concrete and yielding
of steel angles. The eccentricity and slenderness were
also highly influential for load-carrying capacity.
(4) To predict the residual strength (2nd peak load) after
cover-spalling (1st peak load), a simple approach was
proposed on the basis of the strain compatibility
method considering the confinement effect. The
residual strength was determined by local buckling of steel
angles or crushing of confined concrete, whichever is
earlier, and the proposed simple approach gave a good
This research was supported by grants from the National
Natural Science Foundation of China (Research Fund for
International Young Scientists, Grant No. 51650110498),
and the authors are grateful to the authority for the support.
This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unre
stricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original
author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
(Calculation of material properties
for concrete modeling, Saatcioglu and Razvi
1992; Razvi and Saatcioglu 1999)
ec85;u ¼ eco;u þ 0:0018ðk3Þ2.
For unconfined concrete, eco;u ¼ 0:0028
fc0;c ¼ fc0;u þ k1rle;
eco;u 1 þ 5k3
; ec85;c ¼ ec85;u þ 260k3qteco;c
ment efficiencies in the x- and y-directions (rle ¼ k2 rl for
square sections); rl ¼ RðAtftÞ=ðstbcÞ = average confining
pressure in the x- or y-direction; bc = sectional dimension
ft ¼ Et 0:0025 þ 0:04 3 k2qt=fc0;u
= stress in transverse
reinforcement at the peak stress of confined concrete
(ft fyt); At, st, qt ¼
At;y = st bc;x þ bc;y , Et,
and fyt = area, vertical spacing, sectional ratio, elastic
modulus, and yield strength of transverse reinforcement;
k1 ¼ 6:7ðrleÞ 0:17; k2 ¼ 0:15 ðbc=stÞðbc=wiÞ (k2 1:0) in
the x- or y-direction; wi = spacing of laterally supported
longitudinal bars by hoops and cross-ties (ineffective width
between steel angles in the present study); k3 ¼ 40=fc0;u
1:0); k4 ¼ fyt=500 (k4
1:0); and fc;c
ec [ eco;c.
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