Advances in Steel Structures - part 13 potx

This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank
THREE-DIMENSIONAL HYSTERETIC MODELING OF
THIN-WALLED CIRCULAR STEEL COLUMNS
Lizhi Jiang and Yoshiaki Goto
Department of Civil Engineering, Nagoya Institute of Technology,
Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan
ABSTRACT
An empirical hysteretic model is presented to simulate the three-dimensional cyclic behavior of
cantilever-type thin-walled circular steel columns subjected to seismic loading. This steel column is
modeled into a rigid bar with multiple vertical springs at its base. Nonlinear hysteretic behavior of
thin-walled columns is expressed by the springs. As the hysteretic model for the spring, we modify the
Dafalias and Popov's bounding-line assumption in order to take into account the degradation caused
by the local buckling. The material properties for the vertical spring are determined by using curve-
fitting technique, based on the in-plane restoring force-displacement hysteretic relation at the top of the
column obtained by FEM analysis. By properly increasing the number of springs, the homogeneity of
thin-walled circular columns is maintained. Finally this model is used in three-dimensional earthquake
response analysis.
KEYWORDS
Hysteretic model, Three-dimensional behavior, Local buckling effect, Steel, Thin-walled column,
Empirical model, Earthquake response analysis
INTRODUCTION
In the three-dimensional earthquake response analysis for thin-walled steel columns used as elevated
highway piers shown in Fig. 1, FEM analysis using shell elements is the only direct procedure that can
consider both axial force and biaxial bending interaction and local buckling effect. However, it
requires a large amount of computing. Herein, we propose a simple three-dimensional hysteretic model
for thin-walled circular steel columns. To consider the three-dimensional interaction, Aktan And
Pecknold (1974) developed a filament model. However, their model cannot consider the effect of the
local buckling, since they adopt the bilinear relation for the hysteretic model of each filament. The
model we propose herein is alike the filament model but uses fewer springs which simulate the three-
dimensional interaction. As the hysteretic model for each spring, we modify the Dafalias and Popov
(1976) bounding-line model in order to take into account the degradation caused by the local buckling.

The force-displacement relationship for each spring is determined by using curve-fitting technique,
101
102 L. Jiang and Y. Goto
based on the in-plane restoring force-displacement hysteretic relation obtained by the FEM shell
analysis. Liu
et al (1999) also proposed an empirical hysteretic model ,but the application of this
model is restricted to in-plane case. The validity of our model is examined by comparing with the
results of the three-dimensional nonlinear dynamic response analysis using shell element.
Fig. l:Thin-walled steel columns of elevated highways in Japan
BOUNDING-LINE MODEL IN FORCE SPACE
In-plane Hysteretic Behavior of Thin-Walled Circular Steel Columns
From the elastic theory, Timoshenko and Gere (1961), the elastic buckling of columns with circular
section is governed by two structural parameters R, and 3, .
R,
= R. __.ay
X/3( 1-v
2) (1)
t E
~ _ 2L 1 ~-~
(2)
r ~
where R and t are the radius and the thickness, respectively, of the thin-walled circular column; cry is
the yield stress of steel ; E is Young's modulus;v is Poisson's ratio; L is the height of column and r is
the radius of gyration of cross section. In the plastic range, we assume that the hysteretic behavior of
thin-walled circular steel columns is influenced by the axial load ratio
P/Py (Py -Cry,, A and A is the
cross-sectional area) in addition to the two structural parameters
R t and X. As a result of FEM
analysis, hysteretic behavior of thin-walled circular steel columns is classified into three types,
depending on the value

ofR t , as illustrated in Fig. 2 (a), (b) and (c).Herein, the material behavior of
steel is assumed to be represented by the three-surface cyclic plasticity model proposed by Goto et al
(1998). The material constants used for the three-surface model is shown in Table 1.
Considering the sizes as well as the design loads of real columns, three parameters take the values as
0.1 ___
P/Py <_ 0.3,0.06 < R t < 0.12, and 0.2 < 3. < 0.5. These ranges for the three parameters indicate
that the hysteretic behavior of our concern corresponds to that shown in Fig. 2 (b). This hysteretic
behavior is characterized by the gradual strength degradation with the increase of cyclic plastic
deformation.
Three-Dimensional Hysteretic Modeling of Th&-Walled Circular Columns
TABLE 1
THREE-SURFACE MODEL PARAMETERS
Parameter E (Gpa)
Cry
(MPa)
cru(MPa) v
eyp" L/Cry [3
495.0 0.3 0.0183 0.581
SS400 (No.8) 206.0 289.6
103
t ~ H&i/E HmPo.
100 2 0.05 Note
Note: For details see Goto
et al
(1998).
Fig. 2: Classification of hysteretic behavior (P/Py = 0.1, A, = 0.2 )
Modified Bounding-Line Model
Dafalias and Popov (1976) presented a bounding-line constitutive model to express the cyclic
plasticity of metals. We modify this model to express the in-plane force-displacement relation of steel
columns. As shown in Fig. 3, F and X e denote restoring force and plastic horizontal displacement at

the top of the column respectively. XX and YY are bounding-lines. In order to express the strength
degradation under cyclic loading, the gradient K B of the bounding-lines are assumed to be negative,
being different from the original Dafalias-and-Popov model that adopts a positive gradient for the
bounding-lines. The incremental force-displacement relation for the in-plane behavior of steel columns
is expressed as follows, depending on whether the current state belongs to the elastic range or the
plastic range.
(Elastic range) AF = K e ~ AX (3)
(Plastic range)
AF = K e K e / (K e + K e) 9 AX
(4)
where K E is the elastic tangent stiffness and
Kp
is the plastic tangent stiffness. Based on the
bounding-line model,
K p
is give by
di
Kp = K B + H ~ ~
(5)
6in 6
where K B is the slope of the bounding-line; H is the hardening shape parameter; 6 is the distance
from the current force state to the corresponding bound;
6in
is the value of 6 at the initiation of each
loading process. In the elastic range represented by the straight lines OA and CD in Fig. 3, K e is zero;
when the force reaches the bounding-line BC
,K e
becomes the same as K B ;on the curves AB and DE,
K p is expressed by Eq. 5.
104

L. Jiang and Y. Goto
x
Bounding-line
Y
A
>
"'
~in
Y
Fig. 3: Bounding-line model
Empirical Equations to Determine Parameters
Five parameters are included in our hysteretic
model:Fe,Ke,6i,,K B
and H. Among them, two are
elastic parameters:
F e
is the elastic yielding force;K e is the elastic stiffness. The other three are
related to bounding-line model as mentioned in the previous sub-section. From Fig. 2 (b), it is
observed thatF E ,
K E , 6i,
and K B all change with the increase of the plastic deformation. Thus, it is
assumed that these four parameters are extrapolated from their initial values
F e , Ke, 6i,
andKB
following the same rule as
F e = f 9 F e
(6)
K e = f. K e
(7)
~,. = f.o~~ (8)

K B =f~ B
(9)
where f = 1- logo +
Wp/W e 1
C )" We =-2 "Fe ~ X e
is the elastic work.
W e
is the accumulated plastic
work. C is an empirical function given by
C = 37.75 - 33 ~ )~ - 25 ~
P/Py -
125.
R,
(10)
The initial value of the four parameters:
F e ,K e ,(~in ,KB
and H are given by
I
1o
(O'y P (11)
Fe -~'~ -~-)
3EI 1
KE - L 3 ~ (1+ 5.85 ~
(R/L) 2 )
(12)
KB/K E
= (-0.155"
PIPe
+0.1616)+(-0.5085"
P/Py

-0.1317) ~ ~, +(1.06.
P/Py -
2.3) ~
R t
(13)
6i,,/Fe
= (2.7"
P/Py +
0.48) + (-0.12"
P/Py -
0.012)" X + (-22.967 ~
P/Py -
0.95)"
R t
(14)
Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns
105
H = -5.El0 8 9 )~ + 3.E10 8 (15)
where A and I are the cross-sectional area and the second moment of inertia, respectively, of the steel
columns; P is the vertical dead load. The other variables have the same meaning as in Eq. 1 . F E
andK E are directly obtained from elastic
theory;KB/KE, -~i,/FE
and H are so determined by the
least square method that our model best fits the force-displacement relationship obtained by FEM
analysis using shell elements under monotonic loading.
A comparison of the hysteretic loops between the present empirical hysteretic model and the FEM
shell model is shown in Fig. 4 for the steel column with
P/Py
= 0.1,R, = 0.07 and ~, = 0.5. The
present model will yield an acceptable result when applied to the practical design.

Fig. 4: Comparison between FEM model and empirical hysteretic model
MULTIPLE SPRING MODEL FOR THREE-DIMENSIONAL ANALYSIS
Modeling of Thin-Walled Steel Columns
To express the three-dimensional hyteretic behavior, the steel column is modeled into a rigid bar with
multiple vertical springs at its base, as illustrated in Fig. 5. At the column base, no horizontal relative
displacement is assumed to occur.
Rigid body
Multiple springs
y -,,,,
z
<i- x
Y
os n of spring
Fig. 5: Modeling of steel column
Based on the three-dimensional modeling of steel columns, the following incremental force-
displacement relation is obtained.
106
AFr =
AFz
R 2 n
79
c~ 0i)
R
2 n
9 ( .~ k i 9
cos0/9 sin
0 i)
2v
R "
-

~ (~ ~i. cosO,)
L. Jiang and Y. Goto
R 2 n R "
9 i.cosO/.sinOi
-
.cosOi
L ~ T
n
R2 (~~. oR (~k i~
L 2 9 k i 9
sin20i)
L
9 ki L
9 sinOi)
ki
AX
where
AF x ,AFr,AF z
and AX,AY, AZ are force and displacement increments, respectively, at the
top of the columns;
k i
is the tangent stiffness for the ith spring ;
0 i
is the angle that specifies the
location of the ith spring; n is the total number of springs.
The least number of springs that can have three-dimensional interactive effects is four. But this number
of springs can not ensure the homogeneity. Fig. 6 (a) shows the non-homogeneous force-displacement
relations for the column model with four springs under horizontal force directions: 0~176 ~ and
45 ~ . However, if we increase the number of springs, the column comes to exhibit homogeneity as
illustrated in Fig. 6 (b). The least number of springs that is required for homogeneity is 16.

Fig. 6: Homogeneity of multiple spring model
Constitutive Relation for Multiple Springs
The constitutive relation for the multiple springs is determined, based on the in-plane hysteretic model.
From Eq.16, the in-plane force and displacement relation in the X direction is derived as
R 2 n
l~i'X -" "-~" (E ki "
cOS20i)AX (17)
][ a
By comparing Eq.17 with Eq.4, the multiple spring model parameters
FE.,.pri,,g, Kespring, 6i,,.,.p,.i,,g,
KBvr,,g, and
H.,r, ri,,g
can be related as follows to the parameters of the in-plane hysteretic model.
L
FEspring - F e 9 (~, g" a)
(18)
L 2
KEspring Ke . (._RT. g)
(19)
L
6i,.,prZ,g = 6i" (-R " g" a)
(20)
L 2
K Bspring = K. 9 ( RT " g)
(21)
Three-Dimensional Hysteretic Modeling of Thin-Walled Circular Columns
107
L 2
Hspring = H o (-R 5-o g~ a)
(22)

g = 1/'~ cos 2
0 i
and a = 0.87 .
/
where
THREE-DIMENSIONAL EARTHQUAKE RESPONSE ANALYSIS
Steel Column Model
In order to demonstrate the validity of the multiple spring model, a dynamic response analysis is
carried out under the N-S, E-W and U-D components of the Kobe earthquake ground acceleration
recorded by the Japan Meteorological Agency (JMA). Under the same ground acceleration, FEM
analysis using shell elements illustrated in Fig. 7 is also conducted to examine the accuracy of our
model. For the column material property, we adopts the three-surface model with the material
constants summarized in Table 1.
Fig. 7: FEM shell model
Earthquake Response
The results of the earthquake response analysis obtained by the empirical hysteretic model are shown
in Figs. 8-10, in comparison with those obtained by the FEM shell model. Figure 8 illustrates the loci
of the response sway displacement at the top of the column. Figure 9 shows the E-W component of the
Fig. 8: Loci of response sway displacement
108 L. Jiang and Y. Goto
sway displacement history, whereas Fig. 10 shows the hysteresis loops expressed in terms of the force-
displacement relation. From Figs. 8-10; it is confirmed that the empirical hysteretic model can
simulate the three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance.
Fig. 9: Sway displacement history of the column (East-West)
Fig. 10: Comparison of hysteretic force-displacement relation (East-West)
SUMMARY AND CONCLUDING REMARKS
In view of the application to the practical design analysis, a three-dimensional hysteretic model for the
thin-walled circular column is presented. This model is represented by a rigid bar with multiple
vertical springs at its base. These multiple springs are used to consider both the axial force and biaxial
bending interaction and the local buckling effect. The constitutive relation for each spring is

determined by the curve-fitting technique, based on the in-plane hysteretic behavior of the FEM shell
model. In order to examine the validity of the proposed hysteretic model, a three-dimensional
earthquake response analysis is carried out for a steel column by using both the hysteretic model and
the FEM shell model. As a result, it is confirmed that the proposed hysteretic model can simulate the
three-dimensional seismic behavior of the FEM shell model with an acceptable tolerance.
References
Aktan A. E. and Pecknold A. (1974). R/C Column Earthquake Response in Two Dimensions. Journal
of the Structural Division.ASCE.
ST10, 1999-2015.
Dafalias Y. E and Popov E. E (1976). Plastic Internal Variables Formalism of Cyclic Plasticity.
Journal of Applied Mechanics. ASME. 43:12, 645-651.
Goto Y. and Wang Q. Y. (1998). FEM Analysis for Hysteretic Behavior of Thin-Walled Columns.
Journal of Structural Engineering. ASCE. 124:11, 1290-1301.
Liu Q. Y. and Kasai A. (1999). Parameter Identification of Damage-based Hysteretic Model for Pipe-
section Steel Bridge Piers.
Journal of Structural Engineering. JSCE. 45A:3, 53-64.
Shing-Sham L. and George T. W. (1984). Model for Inelastic Biaxial Bending of Concrete Members.
Journal of Structural Engineering ASCE. 110:11, 2563-2584.
Timoshenko S. P. and Gere J. M. (1961).
Theory of Elastic Stability, McGraw-Hill Kogakusha, LTD.
LOCAL BUCKLING OF THIN-WALLED POLYGONAL COLUMNS
SUBJECTED TO AXIAL COMPRESSION OR BENDING
J.G. Teng, S.T. Smith and L.Y. Ngok
Department of Civil and Structural Engineering
The Hong Kong Polytechnic University, Hong Kong, P.R. China
ABSTRACT
Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic
considerations. Limited literature exists, however, on the stability of the component plate elements of
these columns. A finite strip model is used in this paper to investigate the local buckling behaviour and
strength of these columns subject to either axial compression or uniform bending. Cross-sections of

square, pentagonal, hexagonal, heptagonal and octagonal profiles are considered. Elastic local buckling
coefficients are presented for a variety of plate width-to-thickness ratios. It is shown that the
dimensionless buckling stress coefficient is influenced by two parameters: the nature of the applied
loading and the number of sides of the section. The buckling stress coefficient is higher for bent
sections than axially compressed ones, and this difference can be quite significant. Sections with an
odd number of sides have an enhanced buckling capacity over those with an even number of sides,
with pentagonal sections being the strongest under either axial compression or bending.
KEYWORDS
Buckling, stability, columns, finite strip method, local buckling, polygonal sections.
INTRODUCTION
Thin-walled polygonal section columns are a popular form of construction due mainly to aesthetic
considerations. Common polygonal sections include square, pentagonal, hexagonal, heptagonal and
octagonal shapes. These columns are generally subjected to axial and lateral loads. Limited literature
exists on the stability of the component plate elements of these columns. This paper thus considers the
elastic local buckling capacity of polygonal column sections subjected to axial compression or
bending.
The local buckling of thin-walled columns of box sections has been quite extensively investigated.
Few studies on local buckling in polygonal columns, however, are found in the literature. The local
buckling of long polygonal tubes in combined bending and torsion was investigated by Wittrick and
Curzon (1968) using an exact finite strip method. Bulson (1969) undertook a comprehensive test
109