Introduction Steel–concrete composite structures comprised of steel, reinforced concrete, and steel–concrete composite members have widely used for constructing buildings and bridges due
Trang 1Practical nonlinear analysis of steel –concrete composite frames using
fiber–hinge method
a
Faculty of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet St., Dist 10, Ho Chi Minh City, Vietnam
b Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea
a b s t r a c t
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
Afiber–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 endfiber– hinge segments and an interior elastic segment The static condensation method is applied so that the ele-ment comprising of three segele-ments is treated as one general eleele-ment with twelve degrees of freedom The mid-length cross-section of the endfiber 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 accu-racy 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
struc-tural, economic and construction aspects Therefore, extensive
exper-imental 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
increas-ing needs in havincreas-ing a reliable structural analysis program capable of
predicting the second-order inelastic response of steel–concrete
com-posite 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
re-quires a comprehensive analysis procedure that can consider all key
factors influencing the strength of structure and produce results
con-sistent 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,
espe-cially in a time-consuming earthquake-resistant design The degree
of success in predicting the nonlinear load–displacement response
of frame structures significantly depends on how the nonlinear ef-fects to be simulated in numerical modeling
The steel and concrete components can be modeled separately using plate, shell and solid elements of available commercial three-dimensional nonlinearfinite element packages or self-developed pro-grams of researchers and then are assembled together by some con-nection 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 re-finement while the lumped method allows for a significant simplifica-tion The beam–column member in the former is divided into many finite elements and the cross-section of each element is further modeled
byfibers 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
⁎ Corresponding author Tel.: +82 2 3408 3291; fax: +82 2 3408 3332.
E-mail address: sekim@sejong.ac.kr (S.-E KIM).
1
Formerly Adjunct Researcher of Constructional Technology Institute, Sejong
Uni-versity, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea.
0143-974X/$ – see front matter © 2012 Elsevier Ltd All rights reserved.
Contents lists available atSciVerse ScienceDirect
Journal of Constructional Steel Research
Trang 2stress in eachfiber of the steel section can directly be assigned as
con-stant value since thefibers are sufficiently small The solution of the
dis-tributed method can be considered to be relatively accurate and easily
be included the coupling effects among of axial, lateral, and torsion
de-formations However, it is generally recognized that this method is too
computationally intensive and hence usually applicable only for
re-search purposes (e.g., checking and calibrating the accuracy of
simpli-fied 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
time-consuming, 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
ap-plied 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,
es-pecially for steel I-beam and reinforced concrete slab section, is not
always available and accurate for every section Moreover, the
gradu-al reduction in strength of the genergradu-al 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
weak-ness, afiber–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
endfiber–hinge segments and an interior elastic segment to simulate
the inelastic behavior of the material The cross-section at mid-length
of endfiber–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 theflow theory of plasticity This is a good
alter-native 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
exam-ples 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
2 Formulation
2.1 Basic assumptions
The following assumptions are made in the formulation of the
composite beam–column element:
1 All elements are initially straight and prismatic Plane
cross-section 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
4 Reductions of torsional and shear stiffnesses are not considered in thefiber–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 solu-tion time Generally only one element is needed per a physical member
in modeling to accurately capture the P−δ effect Similar to the formu-lation procedure presented by Chen and Lui[11], the incremental force– displacement relationship of the space beam–column element may be expressed as[10]
_P _MyA _MyB _MzA _MzB _T
8
>
>
>
>
>
>
9
>
>
>
>
>
>
¼
EA
ð Þc
0 kiiy kijy 0 0 0
0 kijy kiiy 0 0 0
0 0 0 kiiz kijz 0
0 0 0 kijz kiiz 0
0 0 0 0 0 ð ÞGJ c
L
2 6 6 6 6 6
3 7 7 7 7 7
_δ _θyA _θyB _θzA _θzB _ϕ
8
>
>
>
>
>
>
9
>
>
>
>
>
>
ð1Þ
where _MnA, _MnB, _P , and _T are incremental end moments with respect to
n axis (n = y, z), axial force, and torsion, respectively; _θnA, _θnB, _δ, and _ϕ are incremental joint rotations with respect to n axis, axial displace-ment, and the angle of twist, respectively;
kiin¼ S1nðEInÞc
kijn¼ S2n
EIn
ð Þc
and EA
ð Þc¼Xm i¼1
EIn
ð Þc¼Xm i¼1
GJ
ð Þc¼Xm i¼1
Gi y2i þ z2 i
in which m is the total number offibers divided on the monitored cross-section; Eiand Aiare the tangent modulus of the material and the area of
ithfiber, respectively; yiand ziare the coordinates of ithfiber 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
S1n¼
π ffiffiffiffiffiffiρn p sinπ ffiffiffiffiffiffiρn p
−π2
ρncos π ffiffiffiffiffiffiρn p
2−2 cos π ffiffiffiffiffiffiρn
p
−π ffiffiffiffiffiffiρn p sinπ ffiffiffiffiffiffiρn p
if Pb 0
π2
ρncosh π ffiffiffiffiffiffiρn
p
−π ffiffiffiffiffiffiρn p sinhπ ffiffiffiffiffiffiρn p
2−2 cosh π ffiffiffiffiffiffiρn
p
þ π ffiffiffiffiffiffiρn p sinhπ ffiffiffiffiffiffiρn p
if P > 0
8
>
>
>
>
ð4aÞ
S2n¼
π2
ρn−π ffiffiffiffiffiffiρn p sinπ ffiffiffiffiffiffiρn p
2−2 cos π ffiffiffiffiffiffiρn
p
−π ffiffiffiffiffiffiρn p sinπ ffiffiffiffiffiffiρn p
if Pb 0
πpffiffiffiffiffiρysinh π ffiffiffiffiffiρ
y p
−π2
ρy
2−2 cosh π ffiffiffiffiffiffiρn
p
þ π ffiffiffiffiffiffiρn p sinπ ffiffiffiffiffiffiρn p
if P > 0
8
>
>
>
>
ð4bÞ
whereρ = P/(π2EI/L2) with n = y, z and P is positive in tension
Trang 32.3 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 endfiber 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 manyfibers to track the inelastic
behavior of the section and element (Fig 2) Eachfiber 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 tofibers 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
inte-rior nodes (nodes 3 and 4 inFig 1) must be condensed out to leave
twelve DOFs of two exterior nodes (nodes I and J, also denoted as
nodes 1 and 2 inFig 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
re-duced 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 thefiber–hinge
In a general nonlinear analysis, the element stiffness matrices of the
current step are evaluated based on the state of the system
deter-mined at the end of previous step Once the incrementalfiber strain
of the cross section is evaluated, theflow theory of plasticity is
ap-plied to determine the incrementalfiber stress based on the relevant
uniaxial material stress–strain relationship represented in the
follow-ing section The strain hardenfollow-ing 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
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
de-scribed as following expressions
σ ¼ fc′ 1− 1−ε
ε0
n
where fc′ is the concrete compressive cylinder strength; n is the
expo-nent;ε0is the concrete strain at maximum stress andεuis the
ulti-mate strain For fc′≤50 MPa, n=2 and ε0= 0.002 and these values
are used for all relevant examples presented in this research
2.5 Element stiffness matrix accounting for P−Δ effect The P−Δ effect is the effect of axial load P acting through the rel-ative transverse displacement of the member endsΔ The end forces and displacements used inEq (1)are shown inFig 4(a) The sign convention for the positive directions of element end forces and dis-placements of afinite element is shown inFig 4(b)
Fig 1 Beam–column element comprising of three segments.
bf
Concrete fiber
z
d
bs
ds
Steel fiber
y
Fig 2 The partition of monitored cross-section into fibers.
(a) Steel
(b) Concrete
Trang 4By comparing the twofigures, the equilibrium and kinematic
rela-tionships can be expressed in symbolic form as
_fn
n o
¼ T½ T
_de
n o
where _fn on
andn o_dL are the incremental end force and displacement
vectors of an element and are expressed as
_fn
n oT
¼ rf n1 rn2 rn3 rn4 rn5 rn6 rn7 rn8 rn9 rn10 rn11 rn12g
ð6aÞ _dL
n oT
¼ df 1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12g
ð6bÞ and _fe
n o
and _de
n o
are the incremental end force and displacement vectors inEq (1) [T]6 × 12is a transformation matrix written as
T
½ 612¼
0 0 −1
L 0 1 0 0 0
1
L 0 0 0
0 0 −1L 0 0 0 0 0 1
L 0 1 0
0 1
L 0 0 0 1 0 −1L 0 0 0 0
0 1
L 0 0 0 0 0 −1
L 0 0 0 1
0 0 0 1 0 0 0 0 0 −1 0 0
2
6
6
6
6
6
6
4
3 7 7 7 7 7 7 5 ð7Þ
Using the transformation matrix by equilibrium and kinematic
re-lations, the force–displacement relationship of an element may be
written as
_fn
n o
¼ K½ _dnn oL
ð8Þ [Kn] is the element stiffness matrix expressed as
K
½ ¼ T½ T
K
Eq (9)can be partitioned as
Kn
½ 1212¼ ½ Kn1 ½ Kn2
Kn
½ T
2 ½ Kn3
ð10Þ where
Kn
½ 1¼
0 0 d 0 −e 0
0 0 −e 0 g 0
2 6 6 6 4
3 7 7 7 5
ð11aÞ
Kn
½ 2¼
2 6 6 6 4
3 7 7 7 5
ð11bÞ
Kn
½ 3¼
a 0 0 0 0 0
0 b 0 0 0 −c
0 0 d 0 e 0
0 0 0 f 0 0
0 0 e 0 m 0
0 c 0 0 0 n
2 6 6 6 4
3 7 7 7 5
ð11cÞ
where
a¼EtA
L b¼Ciizþ 2Cijzþ Cjjz
L2 c¼Ciizþ Cijz
L
d¼Ciiyþ 2Cijyþ Cjjy
L2 e¼Ciiyþ Cijy
L f ¼GJ
L
g¼ Ciiy h¼ Ciiz i¼ Cijy j¼ Cijz m¼ Cjjy n¼ Cjjz
ð12Þ
Eq (8)is used to enforce no side-sway in the beam–column mem-ber If the beam–column member is permitted to sway, additional axial and shear forces will be induced in the member These
addition-al axiaddition-al and shear forces due to a member sway to the member end displacements can be related as
_fs
n o
¼ K½ _dsn oL
ð13Þ wheren o_fs , _dn oL
, and [Ks] are incremental end force vector, end dis-placement vector, and the element stiffness matrix They may be written as
_fs
n oT
¼ rf s1 rs2 rs3 rs4 rs5 rs6 rs7 rs8 rs9 rs10 rs11 rs12g
ð14aÞ _dL
n oT
¼ df 1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12g
ð14bÞ
Ks
½ 1212¼ ½ Ks − K½ s
− K½ s T
Ks
½
ð14cÞ where
Ks
½ ¼
0 a −b 0 0 0
a c 0 0 0 0
−b 0 c 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
2 6 6 6 4
3 7 7 7 5
ð15Þ
(a) Forces
(b) Displacements
Fig 4 Element end force and displacement notations.
Trang 5a¼MzAþ MzB
L2 ; b¼MyAþ MyB
L2 ; c¼P
By combiningEqs (8) and (13), we obtain the general beam–column
element force–displacement relationship as
_fL
n o
where
_fL
n o
¼ _fn on
þ _fn os
ð18Þ K
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 commercialfinite
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
compres-sion outside the elastic range following the nonlinear curve of
Eq (28) Compressive stress data are provided as a tabular function of
inelastic strain
3.1 Steel Pinned-Ended Column
The nonlinear analyses are performed for axially compressed steel
pinned-ended column as shown inFig 5to verify the accuracy of the
program in capturing second-order, inelastic, and residual stress
ef-fects The section is W8 × 31 and the yield stress and Young's modulus
of the material are E = 200 GPa andσy= 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 6presents a comparison of buckling loads obtained by the
proposed program's analysis, the above-mentioned Euler's theoretical
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 equa-tion 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 axisλcyof all cases are shown inFig 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
3.2 Continuous composite beam tested by Ansourian 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 plasticityfinite element method to model this beam This beam has two spans 4 m and 5 m long and is loaded by a con-centrated 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
0.0 0.2 0.4 0.6 0.8 1.0
Euler's theoretical solution Fiber hinge element (proposed) CRC curve
Fiber hinge element (proposed) residual stress included
λcy Fig 6 Strength curve of pinned-ended steel column.
P = 200 kN
A A
L = 4 m
A - A IPE 200
800 mm
L = 5 m v
Trang 6studs 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 × 105MPa
Two proposed elements and fourfinite 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
3.3 Composite portal frame
Fig 9shows a steel–concrete composite portal frame comprising of a
steel–concrete composite beam rigidly connected to two steel columns
The compressive cylinder strength of concrete is fc′=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 × 105MPa, and the strain hardening modulus ES= 6 × 103MPa 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 lat-eral 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 topflange area of the steel beam and the corresponding concrete slab area is fully con-strained by using the *TIE function of ABAQUS to simulate a fully com-posite interaction between two components
The load–displacement curves obtained by ABAQUS and the pro-posed programs are shown inFig 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 pro-posed programs for this problem are 48 min and 20 s, respectively This result proves the high computational efficiency of the proposed computer program
0.0
0.2
0.4
0.6
0.8
1.0
Deflection, v (mm)
Experiment, Ansourian (1981) Plastic zone method, Pi et al (2006) Fiber hinge element (proposed)
Fig 8 Load–displacement curves of composite beam CTB1.
A - A
W12x27
1219 mm
P
P = 150 kN
x H = 5 m
A
A u
L = 8 m
Fig 9 Steel–concrete composite portal frame.
0.0 0.3 0.6 0.9 1.2 1.5
Displacement, u (mm)
Shell and solid elements, ABAQUS Fiber hinge element (proposed)
Fig 10 Load–displacement curves of composite portal frame.
Trang 73.4 Steel arch bridge with concrete slab
Fig 11shows 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 × 105MPa and yield
stress of fy= 248 MPa are used for steel material For the concrete
material, the compressive cylinder strength is f′=27.58 MPa and the ul-c
timate strain isεu= 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
inFig 12is used for the bridge tie beams The steel square box section
of 24 × 24 × 1/2 is used for the arch ribs while the wideflange section
of W8 × 10 and W10× 22 are used for the vertical truss members and
the lateral braces of the bridge, respectively.Fig 13shows the design
loads applied to the structure as concentrated vertical and lateral loads
Each member of the bridge is modeled by one proposedfiber–
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
compos-ite 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 ratioλ 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
stiff-ness 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
4 Conclusions
In an effort of reducing computational expense and supplying struc-tural engineers with a reliable and efficient tool in daily engineering de-sign, a practical nonlinear analysis program for predicting nonlinear behavior of steel–concrete composite frame structures is proposed Based on the lumped plasticity concept, thefiber–hinge element com-prised of two exteriorfiber 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 elas-tic segment to capture the second-order effect of the member Using the static condensation algorithm, the element with three segments is trea-ted 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 stor-age 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 con-ducting 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 pro-posed analysis program can be used to evaluate the strength and stabil-ity of steel–concrete composite structures as an integrated system rather than a group of individual members This is very efficient to de-sign the required structure with uniform safe factor and then brings about economic efficiency It can be concluded that the proposed nu-merical procedure is simple but efficient for use in practical design Acknowledgements
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)
References
[1] Baskar K, Shanmugam NE, Thevendran V Finite-element analysis of steel-concrete composite plate girder J Struct Eng 2002;128(9):1158–68.
[2] Barth KE, Wu H Efficient nonlinear finite element modeling of slab on steel stringer bridges Finite Elem Anal Des 2006;42(14–15):1304–13.
W21x101
800 mm
Fig 12 Composite section of tie beams.
(a) Vertical load
(b) Lateral load
30λ
167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ
(Units : kN)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Mid-span displacement ratio, v/L
Bare steel bridge Bridge considering concrete slab
1.123 1.198 Limit Loads
Fig 14 Load–displacement curves of steel arch bridge.
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