DSpace at VNU: Practical nonlinear analysis of steel–concrete composite frames using fiber–hinge method

Journal of Constructional Steel Research 74 (2012) 90–97

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Journal of Constructional Steel Research

Practical nonlinear analysis of steel–concrete composite frames using
fiber–hinge method
Cuong NGO-HUU a, 1, Seung-Eock KIM b,⁎
a
b

Faculty of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet St., Dist. 10, Ho Chi Minh City, Vietnam
Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea

a r t i c l e

i n f o

Article history:
Received 26 July 2011
Accepted 28 February 2012
Available online 28 March 2012
Keywords:
Steel–concrete composite frames
Nonlinear analysis
Fiber–hinge method
Stability functions

a b s t r a c t
A fiber–hinge beam–column element considering geometric and material nonlinearities is proposed for

modeling steel–concrete composite structures. The second-order effects are taken into account in deriving
the formulation of the element by the use of the stability functions. To simulate the inelastic behavior
based on the concentrated plasticity approximation, the proposed element is divided into two end fiber–
hinge segments and an interior elastic segment. The static condensation method is applied so that the element comprising of three segments is treated as one general element with twelve degrees of freedom. The
mid-length cross-section of the end fiber segment is divided into many fibers of which the uniaxial material
stress–strain relationship is monitored during analysis process. The proposed procedure is verified for accuracy and efficiency through comparisons to the results obtained by the ABAQUS structural analysis program
and established results available from the literature and tests through a variety of numerical examples. The
proposed procedure proves to be a reliable and efficient tool for daily use in engineering design of steel
and steel–concrete composite structures.
© 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Steel–concrete composite structures comprised of steel, reinforced
concrete, and steel–concrete composite members have widely used
for constructing buildings and bridges due to their efficiency in structural, economic and construction aspects. Therefore, extensive experimental and theoretical studies have been conducted to provide a
better understanding on the behavior of the composite structure
and its components under applied loading. Together with the more
and more application of the composite structures, there are increasing needs in having a reliable structural analysis program capable of
predicting the second-order inelastic response of steel–concrete composite structures. Recently, as the design profession moves towards a
performance-based approach, the accurate detailed information on
how a structure behaves under different levels of loads is necessary
in evaluation of the expected level of performance. Obviously, this requires a comprehensive analysis procedure that can consider all key
factors influencing the strength of structure and produce results consistent with the current design code requirements with sufficient
accuracy. For daily design purpose, the nonlinear analysis program
should be able to get the reliable results in a minimized time, especially in a time-consuming earthquake-resistant design. The degree
⁎ Corresponding author. Tel.: + 82 2 3408 3291; fax: + 82 2 3408 3332.
E-mail address: (S.-E. KIM).
1
Formerly Adjunct Researcher of Constructional Technology Institute, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea.
0143-974X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jcsr.2012.02.018

of success in predicting the nonlinear load–displacement response
of frame structures significantly depends on how the nonlinear effects to be simulated in numerical modeling.
The steel and concrete components can be modeled separately
using plate, shell and solid elements of available commercial threedimensional nonlinear finite element packages or self-developed programs of researchers and then are assembled together by some connection or interface elements to simulate the shear connectors/
interaction between these components, as recently presented by
Baskar et al. [1] and Barth and Wu [2], among others. This continuum
method can best capture the nonlinear response of the composite
structures and is usually used instead of conducting the high cost
and time-consuming full-scale physical testing. However, in order to
model a complete structure, so many shell, plate, and solid finite
elements must be used and, as a result, it is too time-consuming.
To reduce the modeling and computational expense, “line element”
method has been proposed and it can be classified into distributed and
lumped plasticity approaches based on the degree of refinement used to
represent inelastic behavior. The distributed method uses the highest refinement while the lumped method allows for a significant simplification. The beam–column member in the former is divided into many
finite elements and the cross-section of each element is further modeled
by fibers of which the stress–strain relationships are monitored during
the analysis process, as recently presented by Ayoub and Filippou [3],
Salari and Spacone [4], Pi et al. [5], McKenna et al. [6], among others.
Therefore, this method is able to model the plastification spreading
throughout the cross-section and along the member length. The residual


C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

stress in each fiber of the steel section can directly be assigned as constant value since the fibers are sufficiently small. The solution of the distributed method can be considered to be relatively accurate and easily
be included the coupling effects among of axial, lateral, and torsion deformations. However, it is generally recognized that this method is too
computationally intensive and hence usually applicable only for research purposes (e.g., checking and calibrating the accuracy of simplified inelastic analysis methods, and establishing design charts and

equations) because a very refined discretisation of the structure is
necessary and the numerical integration procedure is relatively timeconsuming, especially for large-scale space structures as normally
encountered in design. Therefore, it is not efficient to apply them in a
daily practical design.
The beam–column member in the lumped method is modeled by
an appropriate method eliminating its further subdivision, and the
plastic hinges representing the plastic interaction between axial
force and the biaxial moments are assumed to be lumped at both
ends of the member. This plastic hinge is usually based on a specific
yield surface and an approximate function to simulate the gradual
yielding of the cross-section. Although this method is less accurate
in comparison with the distributed method, it was shown to be very
simple, fast, and capable of providing results accurate enough for
practical design, as presented by Porter and Powell [7], El-Tawil and
Deierlein [8] and Liu et al. [9]. However, this method is usually applied for nonlinear analysis of frame structures composed of steel,
reinforced concrete and encased composite members because the
yield surface for steel and reinforced concrete composite section, especially for steel I-beam and reinforced concrete slab section, is not
always available and accurate for every section. Moreover, the gradual reduction in strength of the general composite section under
gradual loading is not easy to model.
In this research, to take advantage of computational efficiency of the
common lumped approach and overcome its above-mentioned weakness, a fiber–hinge beam–column element is introduced to model the
steel and steel-composite composite members. This is a development
from the work done by Ngo-Huu and Kim [10] for nonlinear analysis
of steel space structures. The proposed element is divided into two
end fiber–hinge segments and an interior elastic segment to simulate
the inelastic behavior of the material. The cross-section at mid-length
of end fiber–hinge segment is divided into steel and/or concrete fibers
so that the uniaxial stress–strain relationship of cross-sectional fibers
can be monitored during the analysis process based on the relevant
constitutive model and the flow theory of plasticity. This is a good alternative for inelastic representation instead of using the specific yield

surface in usual plastic hinge model. Herein, the stability functions
obtained from the exact buckling solution of a beam–column subjected
to end forces are used to accurately capture the second-order effects.
The nonlinear responses of structures in a variety of numerical examples of steel and steel–concrete composite frames are compared with
the existing exact solutions, the results from the experiments, and
those obtained by the finite element package ABAQUS and plastic
zone analyses to show the reliability and efficiency of the proposed
approach in applying for practical design purpose.

4. Reductions of torsional and shear stiffnesses are not considered in
the fiber–hinge.
5. The connection and bond between member and its components
are perfect. The panel–zone deformation of the beam-to-column
joint is neglected.
2.2. Beam–column element accounting for P − δ Effect
To capture the effect of the axial force acting through the lateral
displacement of the beam–column element relative to its chord (P − δ
effect), the stability functions are used to minimize modeling and solution time. Generally only one element is needed per a physical member
in modeling to accurately capture the P − δ effect. Similar to the formulation procedure presented by Chen and Lui [11], the incremental force–
displacement relationship of the space beam–column element may be
expressed as [10]
2
ðEAÞc
9
8
_ >
P
>
6 L
>

>
>
>
_ >
>
6
M
>
yA >
> 6 0
>
>
= 6 0
< _ >
M yB
6
¼6
_ >
> 6 0
>M
>
zA >
>
6 0
>
>
>
_ >
> 6
>M

>
; 4
: zB >
T_
0

0

0

0

kiiy
kijy
0
0

kijy
kiiy
0
0

0
0
kiiz
kijz

0
0
kijz

kiiz

0

0

0

0

3
9
8
δ_ >
7>
>
>
>
>
> θ_ yA >
>
0 7
>
>
7>
>
>
<
7
0 7 θ_ yB =

7
0 7>
_ >
>
>
> θ zA >
0 7
>
> θ_ >
7>
>
>
zB >
5>
;
:
ðGJ Þc
_ϕ
L
0

ð1Þ

kiin ¼ S1n

ðEI n Þc
L

ð2aÞ

kijn ¼ S2n

ðEI n Þc
L

ð2bÞ

and
m
X

ðEAÞc ¼

ð3aÞ

Ei Ai

i¼1

ðEI n Þc ¼

m
X

2

ð3bÞ

Ei ni Ai

i¼1

ðGJ Þc ¼

m
X



2
2
Gi yi þ zi Ai

ð3cÞ

i¼1

in which m is the total number of fibers divided on the monitored crosssection; Ei and Ai are the tangent modulus of the material and the area of
i th fiber, respectively; yi and zi are the coordinates of i th fiber in the
cross-section; S1n and S2n (n = y, z) are the stability functions with
respect to y and z axes, and are shown as

S1n

The following assumptions are made in the formulation of the
composite beam–column element:
1. All elements are initially straight and prismatic. Plane crosssection remains plane after deformation.
2. Local buckling and lateral–torsional buckling are not considered. All
members are assumed to be fully compact and adequately braced.
3. Large displacements are allowed, but strains are small.

0

_ ,M
_ , P_ , and T_ are incremental end moments with respect to
where M
nA
nB
_ and ϕ_
n axis (n = y, z), axial force, and torsion, respectively; θ_ nA , θ_ nB , δ,
are incremental joint rotations with respect to n axis, axial displacement, and the angle of twist, respectively;

2. Formulation
2.1. Basic assumptions

91

S2n

8 pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ
À pffiffiffiffiffiffiÁ
>
π ρn sin π ρn −π 2 ρn cos π ρn
>
>
À
Á
>
if P b 0
p

p
ffiffiffiffiffi
ffi
ffiÁ
ffiffiffiffiffi
ffi
< 2−2 cos π ρ −π ρ sinÀπ pffiffiffiffiffi
ρn
n
n
¼
À
Á
À
Á
p
p
ffiffiffiffiffi
ffi
p
ffiffiffiffiffi
ffi
ffiffiffiffiffi
ffi
>
π2 ρn cosh π ρn −π ρn sinh π ρn
>
>
À pffiffiffiffiffiffiÁ if P > 0
>

pffiffiffiffiffiffi
ffiÁ
: 2−2 coshÀπ pffiffiffiffiffi
ρn þ π ρn sinh π ρn

ð4aÞ

8
pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ
π2 ρn −π ρn sin π ρn
>
>
>
À pffiffiffiffiffiffiÁ
ffi À pffiffiffiffiffiffiÁ
>
< 2−2 cos π ρ −π pffiffiffiffiffi
ρn sin π ρn
n
¼
pffiffiffiffiffi
pffiffiffiffiffi
>
π ρy sinh π ρy −π2 ρy
>
>
>
À pffiffiffiffiffiffiÁ
:
pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ

2−2 cosh π ρn þ π ρn sin π ρn

ð4bÞ

if P b 0
if P > 0

where ρn = P/(π2EIn/L2) with n = y, z and P is positive in tension.


92

C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

y

2.3. Beam–column element accounting for material nonlinearity
To model the material nonlinearity based on the concentrated
plasticity approximation, the beam–column element is modeled by
two end fiber segments and a middle elastic segment as shown in
Fig. 1. In order to monitor the gradual plastification throughout the
element's cross-section, the cross-section located at the mid-length
of the end segment is divided into many fibers to track the inelastic
behavior of the section and element (Fig. 2). Each fiber is represented
by its area and coordinate location corresponding to its centroid. The
interior segment is assumed to behave elastically similar to the elastic
part in the common plastic hinge. For hot-rolled steel section, the
residual stresses are directly assigned to fibers as the initial stresses.
The ECCS residual stress pattern of I-shape hot-rolled steel section is
used for this study.

Because the hybrid element has three segments with 24 DOFs, a
static condensation method developed by Wilson [12] is applied so
that the element is treated as one element with 12 DOFs normally
found in a general beam–column element. Twelve DOFs of two interior nodes (nodes 3 and 4 in Fig. 1) must be condensed out to leave
twelve DOFs of two exterior nodes (nodes I and J, also denoted as
nodes 1 and 2 in Fig. 1). This reduces the computational time when
assembling the total stiffness matrix and solving the system of linear
equations. In addition, this twelve DOF element is adaptive with the
existing beam–column program so that the coding time can be reduced. However, it also requires that a reverse condensation needs
be performed in order to compute the deformations at the ends of
the segments to evaluate the degree of yielding of the fiber–hinge.
In a general nonlinear analysis, the element stiffness matrices of the
current step are evaluated based on the state of the system determined at the end of previous step. Once the incremental fiber strain
of the cross section is evaluated, the flow theory of plasticity is applied to determine the incremental fiber stress based on the relevant
uniaxial material stress–strain relationship represented in the following section. The strain hardening of the steel material that causes an
increase in member strength is considered. These let the proposed
approach simulate a more realistic behavior of structure than the
common plastic hinge method does.

bs
Concrete fiber
ds
z

d
Steel fiber

bf
Fig. 2. The partition of monitored cross-section into fibers.

2.5. Element stiffness matrix accounting for P − Δ effect
The P − Δ effect is the effect of axial load P acting through the relative transverse displacement of the member ends Δ. The end forces
and displacements used in Eq. (1) are shown in Fig. 4(a). The sign
convention for the positive directions of element end forces and displacements of a finite element is shown in Fig. 4(b).

2.4. Constitutive model of material
The constitutive material models in explicit functions of strain for
steel and concrete recommended by Eurocode-2 [13] are used in this
research as shown in Fig. 3. The tensile strength of concrete is
neglected. The concrete stress–strain relation in compression is described as following expressions

 !
ε n
σ ¼ fc′ 1− 1−
for 0 ≤ ε ≤ ε0
ε0

ð5aÞ

σ ¼ fc′ for ε0 b ε ≤ εu

ð5bÞ

(a) Steel

where f′c is the concrete compressive cylinder strength; n is the exponent; ε0 is the concrete strain at maximum stress and εu is the ultimate strain. For fc′ ≤ 50 MPa, n = 2 and ε0 = 0.002 and these values
are used for all relevant examples presented in this research.

(b) Concrete
Fig. 1. Beam–column element comprising of three segments.

Fig. 3. Constitution models of material.


C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

93

Eq. (9) can be partitioned as
½K n Š12Â12 ¼

½K n Š1
½K n ŠT2

½K n Š2
½K n Š3

!
ð10Þ

where
2

a
60
6
60
½K n Š 1 ¼ 6
60
6

40
0

(a) Forces

2
6
6
6
½K n Š2 ¼ 6
6
6
4

(b) Displacements

By comparing the two figures, the equilibrium and kinematic relationships can be expressed in symbolic form as
ð6aÞ

where

n o
n o
d_ e ¼ ½T Š6Â12 d_ L

ð6bÞ

a¼

n o

n o
where f_n and d_ L are the incremental end force and displacement
vectors of an element and are expressed as
n oT
f_n ¼ f r n1 r n2 r n3 r n4 r n5 r n6 rn7 r n8 r n9 r n10 r n11

r n12 g
ð6aÞ

d3

d4

d5

d6

d7

d8

d9

d10

d11

d12 g
ð6bÞ

n o
n o
and f_e and d_ e are the incremental end force and displacement
vectors in Eq. (1). [T]6 × 12 is a transformation matrix written as
2

½T Š6Â12

6
6
6
6
6
6
¼6
6
6
6
6
6
4

−1 0

0
0 0
1
0 1
0 −
L

1
0 0
0 −
L
1
0
0 0
L
1
0
0 0
L
0
0
1 0

0
0
0
0
0

0 1

0

0

0

0

0

0

0

0
1
L
1
L

0
0
0

1
1 0 −
0 0
L
1
0 0 −
0 0
L
0 0
0
0 −1

0

0

3

0 07
7
7
7
1 07
7
7
7
0 07
7
7
7
0 15

0

T

0
0
0
0
−d 0
0 −f

e
0
0
0

ð11aÞ

0
0
−e
0
i
0

3
0
c7
7
07
7
07
7
05
j

ð11bÞ

3
0
−c 7

7
0 7
7
0 7
7
0 5
n

ð11cÞ

ð12Þ

Eq. (8) is used to enforce no side-sway in the beam–column member. If the beam–column member is permitted to sway, additional
axial and shear forces will be induced in the member. These additional axial and shear forces due to a member sway to the member end
displacements can be related as
n o
n o
f_ s ¼ ½K s Š d_ L

ð13Þ

n on o
where f_ s , d_ L , and [Ks] are incremental end force vector, end displacement vector, and the element stiffness matrix. They may be
written as
r s2

r s3

r s4

r s5

r s6

r s7

r s8

r s9

rs10

r s11

rs12 g
ð14aÞ

ð7Þ
n oT
d_ L ¼ f d1

d2

d3

d4

d5

d6

d7

d8

d9

d10

d11

d12 g
ð14bÞ

0

ð8Þ

[Kn] is the element stiffness matrix expressed as
½K n Š12Â12 ¼ ½T Š6Â12 ½K e Š6Â6 ½T Š6Â12

3
0
c7
7
07
7
07
7
05

h

0
0
−e
0
g
0

C iiz þ 2C ijz þ C jjz
C iiz þ C ijz
Et A
b¼
c¼
L
L
L2
C iiy þ 2C ijy þ C jjy
C iiy þ C ijy
GJ
f
¼
e
¼
d¼
L
L
L2
g ¼ C iiy h ¼ C iiz i ¼ C ijy j ¼ C ijz m ¼ C jjy n ¼ C jjz

n oT
f_ s ¼ f r s1

Using the transformation matrix by equilibrium and kinematic relations, the force–displacement relationship of an element may be
written as
n o
n o
f_n ¼ ½K n Š d_ L

−a 0
0 −b
0
0
0
0
0
0
0
−c

0
0
0
f
0
0

a 0 0 0 0
60 b 0 0 0
6

60 0 d 0 e
½ K n Š3 ¼ 6
60 0 0 f 0
6
40 0 e 0 m
0 c 0 0 0

n o
n o
T
f_n ¼ ½T Š6Â12 f_e

d2

0
0
d
0
−e
0

2

Fig. 4. Element end force and displacement notations.

n oT
d_ L ¼ f d1

0
b

0
0
0
c

ð9Þ

½K s Š12Â12 ¼

½K s Š
−½K s ŠT

−½K s Š
½K s Š

!
ð14cÞ

where
2

0
6 a
6
6 −b
½K s Š ¼ 6
6 0
6
4 0
0

a −b
c
0
0
c
0 0
0 0
0 0

0
0
0
0
0
0

0
0
0
0
0
0

3
0
07
7
07
7

07
7
05
0

ð15Þ


94

C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

and

Euler's theoretical solution

1.0

b¼

MyA þ MyB
L2

;

P
c¼
L

0.8

By combining Eqs. (8) and (13), we obtain the general beam–column
element force–displacement relationship as
n o
n o
f_L ¼ ½K Šlocal d_ L

Fiber hinge element (proposed)

ð16Þ

ð17Þ

CRC curve
Fiber hinge element (proposed)
residual stress included

0.6

P/Py

M þM
a ¼ zA 2 zB ;
L

0.4

where
0.2

n o n o n o
f_L ¼ f_n þ f_ s

ð18Þ

½K Šlocal ¼ ½K n Š þ ½K s Š

ð19Þ

0.0
0.0

1.0

2.0

λcy

3.0

4.0

5.0

Fig. 6. Strength curve of pinned-ended steel column.

3. Numerical Examples
An analysis program developed based on the above-mentioned
formulation is verified for accuracy and efficiency by the comparisons
of its predictions with the experimental test results, available exact

solution, and the results obtained by the use of the commercial finite
element package ABAQUS [14] and the spread-of-plasticity methods.
In the numerical modeling created by the proposed program, each
frame member is modeled as one or two beam–column elements
using the proposed fiber plastic hinge element. The *CONCRETE
DAMAGED PLASTICITY option in ABAQUS is used for modeling
concrete. The *CONCRETE COMPRESSION HARDENING option is used
to defined the stress–strain behavior of concrete in uniaxial compression outside the elastic range following the nonlinear curve of
Eq. (28). Compressive stress data are provided as a tabular function of
inelastic strain.

exact solution, and CRC column curve (Chen and Lui [11]) with a large
range of the column length. Since the proposed element is based on
the stability functions derived from the governing differential equation of beam–column, it is capable of predicting the exact buckling
load of the column by the use of only one element per member in
modeling. Whereas, as stated by Liew et al. [15], the cubic element
in the common finite element method over-predicts the buckling
loads by about 20% if the pinned-ended column is modeled by one
element. The strength curves corresponding to the slenderness
parameter about weak axis λcy of all cases are shown in Fig. 6. It can
be seen that the curves are almost identical for both cases with the
maximum error of about 2%. This example demonstrates the accuracy
and efficiency of the proposed element in predicting the buckling
loads of the column.

3.1. Steel Pinned-Ended Column

3.2. Continuous composite beam tested by Ansourian

The nonlinear analyses are performed for axially compressed steel

pinned-ended column as shown in Fig. 5 to verify the accuracy of the
program in capturing second-order, inelastic, and residual stress effects. The section is W8 × 31 and the yield stress and Young's modulus
of the material are E = 200 GPa and σy = 250 MPa, respectively. The
radius of gyration about weak-axis of the section is ry = 51.2 mm.
Only one proposed element is used to model the column. Two cases
of excluding and including residual stresses are surveyed.
Fig. 6 presents a comparison of buckling loads obtained by the
proposed program's analysis, the above-mentioned Euler's theoretical

Six continuous steel–concrete composite beams tested by Ansourian
[16] are often used as benchmark tests by other researchers. In this
study, the two-span continuous beam indicated as CTB1 by Ansourian
is used to verify the accuracy of the present method. Pi et al. [5] used
the distributed plasticity finite element method to model this beam.
This beam has two spans 4 m and 5 m long and is loaded by a concentrated load at the mid-length of the shorter span as shown in
Fig. 7. The cross-section of CTB1 consisted of an IPE200 steel section
(flanges 8.5 mm× 100 mm, web 183 mm× 6.5 mm) and a concrete
slab 100 mm× 800 mm. The shear connection consisted of 66 welded

P

P = 200 kN
A
v

A
L=5m

800 mm

IPE 200

100 mm

L

W8x31

L=4m

A-A
Fig. 5. Pined-ended column under axially compressed load.

Fig. 7. Continuous composite beam CTB1.


1.0

1.5

0.8

1.2

Load factor

Load factor

C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

0.6

0.4

95

0.9

0.6

Experiment, Ansourian (1981)
0.2

Plastic zone method, Pi et al. (2006)

Shell and solid elements, ABAQUS

0.3

Fiber hinge element (proposed)

Fiber hinge element (proposed)
0.0

0.0
0

10

20

30

40

50

60

Deflection, v (mm)

0

30

60

90

120

150

180

Displacement, u (mm)

Fig. 8. Load–displacement curves of composite beam CTB1.

Fig. 10. Load–displacement curves of composite portal frame.

studs 19 mm× 75 mm, resulting in a connection strength of 150% of the
required strength in positive bending and 160% in negative bending.
Therefore, as mentioned by Ansourian, the effects of slip from the
test were relatively small. Therefore, the interaction between the steel
I-beam and concrete slab can be considered to be fully restrained. The
contribution of rebars into the strength of concrete slab is assumed
to be negligible in this study. The compressive strength of concrete
is fc′ = 30 MPa, the yield strength of steel fy = 277 MPa, and the elastic
modulus of steel E = 2 × 10 5 MPa.
Two proposed elements and four finite elements of Pi et al. [5] are
used to model each member in the continuous beam. As shown in
Fig. 8, the load–displacement curves obtained from the proposed
method achieves a good approximation of the distributed analysis
of Pi et al. [5] while those of both numerical analyses are slightly
different to the experimental curve.

The compressive cylinder strength of concrete is f′c = 16 MPa and the
ultimate strain isεu = 0.00806. For steel material, the yield strength is
fy = 252.4 MPa, the elastic modulus E = 2 × 10 5 MPa, and the strain
hardening modulus ES = 6 × 103 MPa. The beam-section consists of a
W12 × 27 steel section and a concrete slab 102 mm× 1219 mm. The
W12 × 50 section is used for the columns. A concentrated load
P = 150 kN is applied at the mid-length of the beam while another lateral concentrated force with the same value is applied into the top of
the left column. The vertical and lateral loads are proportionally applied
to the structure until the structure is collapsed. To predict the nonlinear
behavior of the structure, the column and beam are modeled by the use
of one and two proposed elements, respectively. For ABAQUS modeling
served for verification purpose, the bare steel frame is modeled by using
5852 quadrilateral shell elements S4R and the concrete slab is modeled

by using 5376 hexahedral solid elements C3D8R. The top flange area of
the steel beam and the corresponding concrete slab area is fully constrained by using the *TIE function of ABAQUS to simulate a fully composite interaction between two components.
The load–displacement curves obtained by ABAQUS and the proposed programs are shown in Fig. 10. It can be seen that the curves
correlate well. With using the same Intel Pentium 2.21 GHz, 3.00 GB
of RAM computer, the computational times of the ABAQUS and proposed programs for this problem are 48 min and 20 s, respectively.
This result proves the high computational efficiency of the proposed
computer program.

3.3. Composite portal frame
Fig. 9 shows a steel–concrete composite portal frame comprising of a
steel–concrete composite beam rigidly connected to two steel columns.
P = 150 kN
A

u

A
4m

W12x50

W12x50

4m

H=5m

P

L=8m

A-A

W12x27

102 mm

1219 mm

Fig. 9. Steel–concrete composite portal frame.

Fig. 11. Geometry and dimension of steel arch bridge.


96

C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97
Limit Loads
1.198

800 mm
1.2

1.123

200 mm

W21x101

Load factor (λ)

1.0
0.8
0.6
0.4
Bare steel bridge

0.2

Bridge considering concrete slab

Fig. 12. Composite section of tie beams.

0.0
0.000

0.001

0.002

0.003

0.004

Mid-span displacement ratio, v/L

3.4. Steel arch bridge with concrete slab

Fig. 14. Load–displacement curves of steel arch bridge.

Fig. 11 shows a steel arch bridge which is 7.32 m (24 ft) wide and

61.0 m (200 ft) long. The elastic modulus of E = 2 × 10 5 MPa and yield
stress of fy = 248 MPa are used for steel material. For the concrete
material, the compressive cylinder strength is f′c = 27.58 MPa and the ultimate strain is εu = 0. 00467. To include the effect of the concrete slab,
the steel–concrete composite section comprised of the steel beam
W21 × 101 and reinforced concrete slab 200 mm× 800 mm as shown
in Fig. 12 is used for the bridge tie beams. The steel square box section
of 24 × 24 × 1/2 is used for the arch ribs while the wide flange section
of W8 × 10 and W10× 22 are used for the vertical truss members and
the lateral braces of the bridge, respectively. Fig. 13 shows the design
loads applied to the structure as concentrated vertical and lateral loads.
Each member of the bridge is modeled by one proposed fiber–
hinge element. In order to evaluate the increase in strength of the
bridge when the composite action of concrete slab is considered
into the load-carrying capacity of the steel tie, two analysis cases
are performed: (1) the bare steel bridge; (2) the bridge with composite section of tie.
The load–displacement curves of the analyses at the mid-span of
middle tie beam in two above-mentioned cases are shown in
Fig. 14. The bare steel arch bridge encountered ultimate state when
the applied load ratio reached 1.123. The system resistance factor of
0.95 is used since the bare steel bridge collapsed by tension yielding
at the vertical truss member. Since the ultimate load ratio λ results
in 1.07 (=1.123 × 0.95) which is greater than 1.0, the member sizes
of the bridge are adequate for strength requirement. It can be seen
that when the concrete slab is considered in the modeling, the stiffness of the bridge is significantly increased. However, the ultimate
load factor of the bridge considering concrete slab slightly increases
6.3% compared to that of the bare steel bridge.

(Units : kN)

4. Conclusions

In an effort of reducing computational expense and supplying structural engineers with a reliable and efficient tool in daily engineering design, a practical nonlinear analysis program for predicting nonlinear
behavior of steel–concrete composite frame structures is proposed.
Based on the lumped plasticity concept, the fiber–hinge element comprised of two exterior fiber segments and one interior elastic segment
is systematically developed so that the gradual reduction in strength
of section and element of steel and steel–concrete composite members
is reasonably captured. The stability functions are used for middle elastic segment to capture the second-order effect of the member. Using the
static condensation algorithm, the element with three segments is treated as general twelve degree-of-freedom beam–column element. This
helps the proposed element is easily adaptive to the existing program
in order to shorten the coding time and, more importantly, reduces
the number of degrees of freedom of the total structure matrix for storage and computational efficiency. The proposed fiber–hinge element
can be considered as the hybrid element which integrates the dominant
characteristics of both common plastic hinge and finite element
methods. As shown in a variety of numerical examples, the proposed
method using only one or two elements per member is capable of conducting a relatively accurate result compared with time-consuming
continuum and distributed plasticity methods that need to model
many elements for one member. For large-scale structures with many
elements and degrees of freedom in numerical modeling as normally
encountered in a real design, this efficiency is really significant. The proposed analysis program can be used to evaluate the strength and stability of steel–concrete composite structures as an integrated system
rather than a group of individual members. This is very efficient to design the required structure with uniform safe factor and then brings
about economic efficiency. It can be concluded that the proposed numerical procedure is simple but efficient for use in practical design.
Acknowledgements

167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ

7.32 m

(a) Vertical load

30λ 30λ 30λ 30λ 30λ 30λ 30λ 30λ 30λ

(b) Lateral load
Fig. 13. Load conditions of steel arch bridge.

This work was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MEST) (No.
2011-0030847), and the Human Resources Development of the Korea
Institute of Energy Technology Evaluation and Planning (KETEP) grant
funded by the Korea government Ministry of Knowledge Economy
(No. 20104010100520).
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