Advances in Steel Structures - part 47 pptx

440
Z. Wang and Y. Zhen
Figure 1. The comparison between theoretical calculation and the test result
Research on the Hysteretic Behavior of High Strength Concrete
441
Figure 2. The comparison between theoretical calculation and the test result in reference 4
Figure 3. The comparison between theoretical calculation and the test result in reference
5
THE FORCE-DISPLACEMENT HYSTERETIC LOOP CHARACTERISTIC
We can find some characteristics of force-displacement hysteretic loop of high strength concrete filled
steel tubular members from the theoretical analysis and experiment research result.
a) The shape of hysteretic loop is closed to that of steel member under the condition of without local
buckle. And it is also analogous to the loop of general concrete filled steel tube member.
442
Z. Wang and Y. Zhen
b) No matter how the parameters change, the hysteretic loop has great plumpness and no pinched or
reduced phenomenon appear.
4.CONCLUSION
The calculation method in this paper has its new characteristics on how to select constitutive
relationship and construct the model of finite element. On the basis of theoretical analysis and
experiment study, the characteristic of force-displacement hysteretic loop of HCFST under
compression and bending are discussed.
From above, I think the following should be further studied:
a) The basic property of polygon HCFST should be studied by making use of programme in this
paper.
b) The property of the member of eccentric compression should be studied by making use of
calculating method in this paper.
c) The lateral force resisting property of short column of HCFST should be studied considering of
shear deflection.
REFERENCE
l.Shantong Zhong.(1994). Concrete filled steel tubular structures. Heilongjiang science and

technology press,.
2.Linhai Han.(1996) Mechanics of concrete filled steel tubular. Dalian science and engineering
university press.
3.Yonghui Zhen.(1998). The hysteretic behavior studies of high strength concrete filled steel
tububular members subjected to compression and bending. Master thesis of HUAE.
4.Weibo Yan.(1998). Theoretical analysis and experimental research for the hysteretic behaviors of
high strength concrete filled steel tubular beam-columns. Master thesis of HUAE.
5.Yongqing Tu.(1994). The hystersis behavior studies of concrete filled steel tubular membersw
subjected to compression and bending. Doctor thesis of HUAE.
DESIGN OF COMPOSITE COLUMNS OF ARBITRARY CROSS-
SECTION SUBJECT TO BIAXIAL BENDING
S. F. Chen 1, j. G. Teng 2 and S. L. Chan 2
1 Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China
2 Department of Civil and Structural Engineering,
The Hong Kong Polytechnic University, Hong Kong, China
ABSTRACT
In this paper, an iterative Quasi-Newton procedure based on the Regula-Falsi numerical scheme is
proposed for the rapid design of short concrete-encased composite columns of arbitrary cross-section
subjected to biaxial bending. The stress resultants of the concrete are evaluated by integrating the
concrete stress-strain curve over the compression zone, while the stress resultants of the encased
structural steel and the steel reinforcing bars are obtained using the fiber element method. A
particularly important feature of the present method is the use of the plastic centroidal axes of the
cross-section as the reference axes of loading in the iterative solution process. This ensures the
convergence of the solution process for all cross-sectional conditions. Numerical examples are
presented to demonstrate the validity, accuracy and capacity of the proposed method.
KEYWORDS
Composite Columns, Arbitrary Cross-Sections, Irregular Cross-Sections, Structural Design, Biaxial
Bending
INTRODUCTION
Composite steel-concrete construction has been widely used in many structures such as buildings and

bridges. The concrete-encased composite column is one of the common composite structural elements.
Many studies have examined the behaviour and strength of biaxially loaded composite columns of
doubly-symmetric cross-sections. Several researchers, including Johnson and Smith (1980), Lachance
(1982) and Roik and Bergmann (1984) proposed simple methods for the analysis and design of
rectangular composite columns under biaxial loading. E1-Tawil et al. (1995) developed an iterative
computer method for biaxial bending of encased composite columns using the fiber element method
and generated numerical results to evaluate the uni- and biaxial bending strengths of composite
columns predicted by ACI-318 (1992) and AISC-LRFD (1993) provisions. Munoz and Hsu (1997)
proposed a generalized interaction equation for the analysis and design of biaxially loaded square and
443
444
S.F. Chen et al.
rectangular columns. They compared their results with test results of many columns and predictions of
the current ACI (1992) and AISC (Manual 1986) design methods.
In the design of building comer columns, comer walls, and core walls, irregular cross-sections or
regular cross-sections with asymmetrically placed structural steel and/or steel reinforcement are often
used to suit irregular plan layouts and/or eccentric load requirements. Little work has been carried out
on composite columns of such irregular sections. Design rules for these columns also do not exist in
design codes such as ACI-318 (1992) and EuroCode 4 (1994). Roik and Bergmann (1990) appears to
have presented the only study on the design of rectangular composite section with an asymmetrically
placed steel section. They proposed an approximate design method, which is a simple modification of
the design approach for composite columns with a doubly symmetrical cross-section given in
Eurocode 4 (1984). Only sections with the structural steel mono-symmetrically placed were
considered. Rotter (1985) presented a moment-curvature analysis of arbitrary sections subject to axial
load and biaxial bending using Green's theorem in integration which can be used to analyze composite
steel-concrete columns. The emphasis of his study is on predicting the moment-curvature relationship
rather than the design of such sections.
This paper provides a brief description of a general iterative computer method for the rapid design and
analysis of arbitrarily shaped concrete-encased composite columns with arbitrarily distributed
structural steel and steel reinforcement subjected to biaxial bending. A detailed presentation of the

method is given in Chen et al. (1999). The method employs the iterative Quasi-Newton procedure
within the Regula-Falsi numerical scheme. The stress resultants of the concrete are evaluated by
integrating the concrete stress-strain curve over the compression zone, while those of structural steel
and reinforcement are obtained using the fiber element method, in which the steel sections are
discretized into small areas (fibers). Numerical examples are presented to demonstrate the validity,
accuracy and capability of the proposed method.
REFERENCE LOADING AXES
For any cross-section under biaxial loading, the exact location of the neutral axis is determined by two
parameters: the orientation
On
and the depth
dn
(Figure 1). The Quasi-Newton method has been adopted
for the solution of 0 n and dn and found to be effective in many studies (e.g. Brondum-Nielsen, 1985;
Yen, 1991). When dealing with irregular cross-sections, especially those with the arrangement of
structural steel and reinforcement being strongly eccentric, convergence of the iterative process cannot
be guaranteed. Indeed, when such a column is subjected to an axial load with magnitude approaching
the axial load capacity under pure compression, the origin of the loading axes may fall outside the iso-
load contour if this origin is located at the geometric centroid of the cross-section as usual (Yau et al.,
1993). As a result, the inclination of the resultant bending moment resistance
O~m
may change in a
range less than 2re when On varies from 0 to 2~r, resulting in non-uniqueness or non-existence of the
solution of On (Yau et al., 1993). In this paper, this difficulty is overcome by using the plastic centroid
as the origin of the reference loading axes. By taking the plastic centroidal axes as the reference
loading axes, the existence and uniqueness of the solution of
On
are always ensured and the
convergence of the iterative solution process is guaranteed.
For an arbitrary composite cross-section, the plastic centroid may be determined as follows (Roik and

Bergmann, 1990)
XcAcf~c/rc +XsA~L/r, +XrArfy/7"r YcAcfc~/rc +Y~A~L/r~ +YrArfy/7"r
= , r.c= Acfcc/rc+A~L/r~+Arfy/rr
(1)
Ypc AcLc/rc+AsL/r~+A~fy/rr
Design of Composite Columns of Arbitrary Cross-Section
445
where
Ac, Ar and As
are the total areas of concrete, reinforcing bars and structural steel, respectively;
fcc, fy and fi are the respective specified strengths according to design codes such as Eurocode 4
(1994); Yc, Yr and 7~ are the corresponding partial safety factors,
Xc, Yc, Xr, Yr. Xs
and Ys are the
respective centroid coordinates in the global
XCY
system.
v~ T"_~ \ vmax \dn ~''-1~- \1
(a) (b) (c)
Figure 1: Arbitrarily shaped cross-section: (a) Cross-section consisting of several regions; (b) Strain
distribution; (c) Stress block for concrete in compression
CROSS-SECTIONAL DESIGN
Basic Assumptions
The proposed design method is based on the following basic assumptions:
(1) Plane sections before deformation remain plane after deformation. Consequently, the strain at
any point of the cross-section is proportional to its perpendicular distance from the neutral axis.
(2) The cross-section reaches its failure limit state when the strain of the extreme fiber of the
concrete in compression attains the maximum strain
gcu.
(3) The stress-strain relationship of concrete in compression is represented by a parabola and then a

horizontal line:
(c f~22/(~<~0)and
crc=fic(CO<_C <-Ccu)
(2)
Crc=fcc
2~oo-Co J
The structural steel and the steel reinforcing bars are assumed to be elastic-perfectly plastic.
(4) Tensile strength of concrete is neglected.
Stress Resultants in the Cross-section
The cross-section may assume any shape with multiple openings. The entire section may be divided
into several regions if necessary (Figure 1 a). For convenience of calculation, each region is treated as
the superposition of a solid section occupying the entire area of the region and a number of negative
sub-sections representing the openings (Figure 1 a). Consequently, the entire section consists of
ns
solid
subsections (i.e. regions completed filled with structural materials) and
no
negative subsections (i.e.
446
S.F. Chen et al.
openings). All subsections may be arbitrarily polygonal. The
xoy
coordinate system represents the
reference loading axes with its orgin as determined by Eqn. 1. The
uov
system is related to the
xoy
system by a rotational transformation with the u-axis being parallel to the neutral axis.
For the ith subsection (either solid or negative), the coordinates of the jth vertex in the
uov

coordinate
system are related to those in the
xoy
system by
uy=-xjcOS On+yjsin O., vj=yscOS On-Xjsin O.
(3)
where
On
is the orientation of the neutral axis.
In order to evaluate the stress resultants of the concrete, the intersection points of a subsection with the
neutral axis are first determined. These intersection points together with the vertices of the subsection
above the neutral axis form the compression zone of the subsection (Figure 1). The stress resultants of
the concrete can then be found by integrating the concrete stress-strain curve (Figures l b and 1 c) over
this polygonal compression zone as given in Eqn. 4, assuming that the vertices of the zone are
numbered sequentially (either clockwisely or anti-clockwisely)
It
Nzci =]Pzcil = ICrc dud~
IJ=l
uj o
n c Uj+l v(u) n c Uj+l v(u)
, Muci = PiZ
I I [-ere (~ + vn)]dud~' Mvc i = PiE f f~
(4)
j=l
uj 0
j=l
uj 0
where
nc
is the total number of vertices of the compression zone; P

(u)=v(u)-v.
is the linear equation
of the boundary line, with v. being the v-coordinate of the neutral axis; pF1 when
P~ci
>0, and p~ -1
when
Pzci<O; Unc+l Ul
and
Vnc+l V 1.
The total stress resultants contributed by the concrete of the whole cross-section are then given by a
summation over all subsections
n s +n o n s +n o n s +n o
Nzc
= ZciNzci,
Muc=
ZciMuci,
Mvc=
ZciMvci
(5)
i=1 i=1 i=1
in which c~l for a solid subsection and cu -1 for a negative subsection. The bending moments of the
concrete about the x- and y-axes can be easily obtained by coordinate transformation
Mxc=MuccOS O Mvcsin 0~, Myc=Mucsin O.+MvccOS O. (6)
The fiber element method (Mirza and Skrabek, 1991) is used to calculate the stress resultants carried
by the structural steel and the steel reinforcement. The steel section is subdivided into small areas
referred to as fiber elements and the reinforcing bars are treated as individual fibers. Both the structural
steel and the reinforcing bars are assumed to be elastic-perfectly plastic. The stress resultants of the
whole composite cross-section, axial force and bending moments about the x- and y- axes, can then be
written as
mr ms

N z =Nzc+~-'~croAo+ff'o',jAsj
j=l j=l
mr ms
M x = Mxc - ~"
(O'rj
-Crcj)ArjYrj - ~
(o',j
-Crcj)Asjysj
(7)
j=l j=l
mr ms
My
= My c + Z ( O'rj
O'cj ) mrj X rj +
Z ( O'sj
O'cj ) Asj X sj
j=l j=l
Design of Composite Columns of Arbitrary Cross-Section
447
where
mr
and ms are the numbers of reinforcing bars and steel fibers into which the structural steel is
discretized, Crrj, Crsj and ere are the stresses of the reinforcement, the structural steel and the concrete at
the center of thejth bar or steel fiber. If
vj Vn
<0, Crcj =0.
Iterative Solution Procedure
For a given composite cross-section subjected to an axial load
Nzd
at o and bending moments

Mxd and
Myd
about the x- and y-axes respectively (Figure 1), the depth and orientation of the neutral axis
dn and
On
can be determined by the following iterative procedure:
(1) Initial values of
On and d,
are first specified, for which the axial force
Nz
is calculated using the
first expression of Eqn. 7.
(2) The calculated axial force
Nz
is compared to the design value
Nza
and iteratively adjusted using
the following equation until
Nz
is equal to
Nzd
with a given tolerance:
dn '-dn
dn, k = d n + (Nzd - Nz)
(8)
N~'-Nz
in which
Nz' and Nz are
the axial force capacities calculated with the neutral axis depths
dn'

and
dn
respectively, with
Nz'
being greater than the design value
Nzd and Nz
being smaller than
Nzd.
(3) The bending moments
Mx and My are
found using the second and third expressions of Eqn. 7,
and then the angle
Ctm=arctg(My/Mx)
is determined.
(4) The value of
am
is compared to the design value
ama=arctg(Myd/Mxd)
and iteratively adjusted
using the following equation:
O,'-On
On, k = O n + (Ctmd am)
(9)
a m t a m
in which am' and
am
are the inclination of the resultant bending moment calculated with the
neutral axis orientations 0,,' and 0,, respectively, with
am'
being greater than the design value

Ctnd,
and
am
being smaller than
Ctmd.
Steps (1) to (3) are repeated until
O~m
and
amd
are identical within
a given tolerance.
(5) If the task is to check the adequacy of an existing design, the calculated resultant bending
moment
Mr
is compared to the design
value
Mra~(Mxd2+My2) 1/2.
If
Mr
> Mrd,
the cross-section is
adequate, otherwise the structural steel and/or reinforcing bars should be increased and/or the
section enlarged.
(6) If the task is to design a section, all structural parameters except the reinforcing bars are first
specified. Only steel bars of identical properties and size may be used, and the bar diameter can
be determined iteratively starting from an initial assumed value. For any value of the bar
diameter, the corresponding resultant bending moment capacity can be found and the bar
diameter is iteratively adjusted using the following equation:
~bk=~b + r
(Mrd_Mr)

(10)
Mr'-M r
where
Mr'
and Mr are the resultant bending moments resisted by the section with bar diameters
~b' and r respectively, with
Mr'
being greater than the design value
Mrd
and
Mr
being smaller
than
Mrd.
The required bar diameter is found when Mr is equal to
Mra
within a given tolerance.
448
NUMERICAL EXAMPLES
S.F. Chen et al.
A computer program has been developed based on the method presented in this paper. Two numerical
examples are presented below to demonstrate the validity, accuracy and capability of the proposed
method. Further numerical examples can be found in Chen et al. (1999). Tolerances used for the axial
load, the inclination of resultant bending moment and the resultant bending moment are 10 -5, 10 -4 and
10
-5 respectively. The larger tolerance for the inclination of resultant moment was used as it was found
to converge more slowly than the other two parameters.
Rectangular Columns with Asymmetrically Placed Steel H-Section
Two rectangular cross-sections with asymmetrically placed structural steel are shown in Figure 2. The
maximum load-carrying capacities of the cross-sections under uniaxial loading at different

eccentricities were determined in tests by Roik and Bergmann (1990). The cube strength of concrete
and yield stresses of structural steel and reinforcing bars are listed in Table 1.
Figure 2: Rectangular cross-sections with asymmetrically placed structural steel
TABLE 1
MATERIAL PROPERTIES (ROIK AND BERGMANN,
1990)
Specimen Cross-
Section
Vll, V12, V13 V1
V21, V22, V23 V2
Lk fsflange fs.web A,test _
(N/ram 2 ) (N/mm ~) (N/mm 2 ) fN/mm ~ )
37.4 206 220 420
37.4 255 239 420
TABLE 2
LOAD CARRYING CAPACITIES AND COMPARISON WITH TEST RESULTS
Specimen er(mm)
Nz.comp(kN) Nz.test(KN) (Nz.comp-Nz.test)/Nz.test
VII 0 3608 3617 -0.25%
V 12 -40 2654 2825 -6.05%
V13 100 1937 1800 7.61%
V21 0 2880 2654 8.52%
V22 -40 2107 1998 5.46%
V23 100 2036 1706 19.34%
Design of Composite Columns of Arbitrary Cross-Section
449
In order to compare the present results with the test results, all material partial safety factors were
taken to be unity in the analysis. The stress-strain curve from Eurocode 4 (1994) for concrete was
used, that is, fcc = 0.85fck, c0 = 0.002 and ecu=0.0035 in Eqn. 2. Table 2 gives the computed load-
carrying capacities of the six specimens and their comparison with the test results, where er is the load

eccentricity in the Y-direction with reference to the geometric centroid of the cross-section (Figure 2).
It is seen that the computed values agree closely with the test results. This demonstrates the accuracy
and validity of the proposed numerical method.
Column of Asymmetric Polygonal Cross-Section
A composite column cross-section, as shown in Figure 3, is subject to the design loads as follows:
Nzd = 4320.5 kN, Mxd = 950.15 kNm, Mrd = -577.49 kNm
in which X and Y are the geometric centroidal axes of the cross-section. The size and layout of the
encased structural steel and the distribution of the steel reinforcing bars are shown in Figure 3. The
reinforcing bars are assumed to have the same diameter of 12 mm. The task here is to check if the
cross-section has an adequate capacity to carry the given design loads or to determine the required bar
diameter if this section is inadequate.
The stress-strain curve of Eqn. 2 for concrete (Eurocode 4, 1994) is used in the calculation, with
fcc=O.85fck/Yc, c0=0.002 and 6cu=0.0035. The specified strengths and safety factors are taken as follows:
fck=30
N/mm2,fs=355N/mm2,fy=460 N/mm2; yc=l.5, ?~=1.1 and yr=l.15
(1) Checking the adequacy of the pre-defined cross-
section
Coordinates of the plastic centroid o of the pre-
defined cross-section (Figure 3):
Xpt =-25.962 mm, Ypt =-27.694 mm
Design loads with reference to the plastic centroidal
axis system
xoy (Figure 3):
Nza = 4320.5 kN, area = 150.739 ~ Mra-=951.966 kNm
Calculated load carrying capacity of the cross-
section:
Nz = 4320.5 kN, am = 150.739 ~ Mr=774.575 kNm
As
Mr<Mra, the pre-defined cross-section is
inadequate.

Figure 3: Cross-section of a column
(2) Deisgn of the required bar diameter
The program was instructed to find the required bar diameter for the corner column to achieve an
adeqaute resistance for the design loads. The required bar diameter was found to be ~eq=20mm, with
which the cross-section has a load carrying capacity of Nz=4320.5kN, Ctm=150.739 ~ and
Mr = 951.972
kNm.
CONCLUSIONS
An iterative numerical method for the rapid design of short biaxially loaded composite columns of
arbitrary cross-section has been presented in this paper. In the proposed method, the plastic centroidal
axes of the cross-section are taken as the reference loading axes, which ensures the uniqueness and