Seismic behavior of steel rigid frame with imperfect brace members

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 6, Issue 1, January (2015), pp. 113-126 © IAEME
113
SEISMIC BEHAVIOR OF STEEL RIGID FRAME WITH
IMPERFECT BRACE MEMBERS
Hamid Afzali 1
, Toshitaka Yamao2
1, 2
Graduate School of Science and Technology, Kumamoto University, Japan
ABSTRACT
Model of a steel rigid frame made of thin-walled box section with existence of I-section brace
member with initial overall and local imperfection adopted to investigate buckling effects on steel
structural behavior as it was subjected to earthquake excitation. In order to take into account of the
influence of local deflections on structural response, shell elements were employed to model brace
member as well as base columns. Cross sections components with relatively high amplitude of
buckling parameters were considered in different case studies to make it susceptible to develop local
deflection. Beam elements were also utilized to develop models with the same specification. FEM
method applied to conduct nonlinear time history analysis using earthquake record in in-plane and
out-of-plane direction. Seismic response of both shell element model and beam element model were
obtained and compared to investigate the effect of local deformation on seismic behavior of the
structure. It was found that in case of applying earthquake record in longitudinal direction of the
structural frame, due to ignoring local deflections beam element model is not sufficient to present
maximum response for structural case studies made of components with higher buckling parameters.
Buckling deformations were observed and discussed based on obtained results in case of applying
earthquake records in transverse direction.
Keywords: Steel rigid frame, seismic behavior, time history analysis, initial imperfection, buckling
effect
I. INTRODUCTION
Studying seismic behavior of rigid frames with bracings composed of structural members
with relatively thin-walled cross sections members is important since it may be used as part of steel
arch bridges which are frequently subjected to ground motions. When faced with the risk of
instability in thin component plates during severe earthquakes, conventional approach of applying
FEM method using beam elements in analysis procedure and designing of earthquake resistant steel
structures seems to be inadequate to show seismic behavior of the structure [1] [2] [3]. Development
of a model using shell finite element for structures made of built-up cross section with thin plate
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 6, Issue 1, January (2015), pp. 113-126
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114
components helps to consider the effects of local behavior in seismic response. Buckling of a
structural compressive member has encouraged many researches to work on this subject up to now.
Employing shell elements enables us to observe local deformations and its effect on structural
resistance deterioration.
Investigating buckling effects in members of civil structures is an extensive research field.
However, in most of previous works the main concern was buckling behavior of a specimen under
compressive loading such as axial force or bending moment. Many researchers were interested in
discussing compressive structural members alone rather than its influence on whole structure and
other members. For instance, potential interaction between different buckling modes have considered
as important research subject as well as efforts to present a theoretical design equations for buckling
ultimate load. Closed formed prediction of elastic local buckling and distortional buckling based on
interaction of connected elements is presented and examined for lipped channel and Z sections [4].
Sensitivity of compressive capacity of plates with geometric imperfections [5] and sensitivity of
buckling collapse depending on type of shell elements employed in making a model as well as the
density of mesh generation were also examined by researchers. Different software packages used to
make finite element model and finally results obtained by different solvers were compared [6].
Ductility of different cross sections such as steel box section and I-beam sections when they are
subjected to axial load were explored [7] [8]. It is necessary to explore how buckling zones may
grow through structural members under regular loading, and how it affects failure mechanism.
In this paper the numerical finite element model of total structure was provided to study
effect of buckling behavior of more than one member on structural response. This enables us to
observe how buckling effects in a particular part of structure may affect resistance degradation in
whole structural system. Obviously structural member specifications play a major role in forming
local deflections; therefore there is a necessity to explain how variation on a specific parameter in a
structural member contributes to total behavior of the structure. Here target structure was steel rigid
frame with inverted V shape bracing. Effect of slenderness of brace member on behavior of structure
was investigated as well. This structure may be a part of an arch steel bridge. In order to take account
of local behavior in braces and base columns, finite element model was adopted employing shell
elements in targeted zones.
II. LAYOUT OF ANALYSIS
2.1 Analytical model
Target structure was rigid frame with inverted "V" shape brace as illustrated in Fig. 1(a). Fig.
1(b) and Fig. 1(c) are box section for rigid frame, and I-section beam used for brace members. In Fig.
1(b), W=0.30 m denotes side length of the box section, and the thickness of the component plate is
t=0.0064 m.
(a) Rigid frame with brace (b) Sec. 1-1 (c) Sec. 2-2
Figure 1: Geometry and cross sections in structure of rigid frame with brace member
W
W
t
H
W
B
B
Bf
Hw
tw
tf

115
“B” represents total width and height of the I-section. The parameters tw, tf and denote the
thickness of the web and flange, respectively. Constant vertical load of 5 percent of axial yield
capacity of column section imposed on two top corners of the rigid frame in order to consider super
structural load. The material is assumed to be SM400 steel (JIS). The yield stress σy is 235MPa;
Young’s modulus E is 200GPa, and Poisson’s ratio is ν = 0.3. Other specifications are explained in
Table 1. Plot of strain-stress curve is shown in Fig. 2.
Figure 2: Strain-stress curve of material
Aiming to study buckling effects in structures with members made of thin plate components,
width-to-thickness parameter of cross sections was considered and defined as following equations.
Equations (1)-(3) show width-to-thickness ratio for flange, web in I-shape beam, and for side
plate in box section, respectively. Members with higher values of width-to-thickness parameter are
deemed thin-walled sections. σy is yield stress and E is modulus of elasticity. Buckling coefficient
"k" is assumed equal to 4.0 for double-sided stiffened plates such as web in I-shape section and side
plates of the square shape section. This parameter is set to 0.425 for flanges in I-beam sections. Rigid
frame section properties are the same in all case studies. For plates of box-section R=0.8 based on
Equation (3). Aiming to consider probable local deformations in rigid frame, cross section with
relatively slender plates determined for box section. According to JSHB [9], allowable design
compressive stress is decreased for component plates of this section due to considering local
buckling effects under service loading. Buckling parameter amplitude for plates of I-beam section
ranged between 0.6, 0.8, and 1.0 to represent sections made of normal, relatively thin, and thin
plates. Nine specimens with I-beam sections are explained in Table 1. “L” stands for length of the
brace member and “r “is radius of gyration of the section.
0
100
200
300
400
0 0.02 0.04 0.06 0.08 0.1 0.12
σ(N/mm²)
ε
SM400
2
2
)1(12
π
νσ
kEt
B
R
y
f
f
f
−
= (1)
2
2
)1(12
π
νσ
kEt
H
R
y
w
w
w
−
= (2)
2
2
)1(12
π
νσ
kEt
H
R y −
= (3)

116
Table 1: Nine specimen of brace members with I-section
Specimen of brace member B (m) tf (m) tw (m) L/r Rf Rw
100-6-6 0.092 0.0041 0.0025 100 0.6 0.6
100-8-8 0.093 0.0031 0.0019 100 0.8 0.8
100-10-10 0.093 0.0025 0.0016 100 1.0 1.0
90-6-6 0.103 0.0046 0.0028 90 0.6 0.6
90-8-8 0.103 0.0035 0.0022 90 0.8 0.8
90-10-10 0.103 0.0028 0.0018 90 1.0 1.0
80-6-6 0.115 0.0052 0.0032 80 0.6 0.6
80-8-8 0.116 0.0039 0.0024 80 0.8 0.8
80-10-10 0.116 0.0032 0.0020 80 1.0 1.0
2.2 Numerical model
Finite element software package of ABAQUS program [10] were employed to develop a
model and demonstrating the seismic behavior of the structure. For the purpose of approximation of
local deformations, as depicted in Fig. 3(a) shell model were employed in base columns and 83% of
brace member length as regions with large internal forces rather than other parts of the structure, and
most susceptible to local buckling effects.
(a) Shell model (b) Beam model
(c) Connecting beam element to the shell zone using rigid plates
Figure 3: Numerical model
Shell element
Super-structural load
Fixed support Fixed support
Fixed support Fixed support
Beam element

117
Fig. 3(b) depicts the model made of all beam models. Responses of beam model were
compared with that of shell model. In order to provide connectivity between beam elements and shell
elements, end side of the shell element zone and beam zone were connected by rigid plates as shown
in Fig. 3(c). Four node shell element S4R with reduced integration scheme were applied to model
cross sections in base column and brace member. This element is robust and avoids shear locking
that makes it appropriate for wide range of applications. The rigid frame members excluding base
column were made of beam elements B31. These elements are also capable of taking account of
shear deflections. Brace members are assumed to join to rigid frame using pin connections.
Translational degrees of freedom are fixed at the base. Time history analyses were conducted in
longitudinal and transverse direction proportional to the structure.
2.3 Applying initial Crookedness
Thin plate components of built-up sections are not completely flat. Applying loading system
may result in local deformations. Small initial crookedness and slenderness of the plates could lead
in transverse deformations under compression. The axial load may generated by compression or
bending moment. Crookedness influence local buckling behavior while the structure is subjected to
external pressure also has been concerned in many researches. Imperfections may appear in two
different types: geometric and stress. There is no verified theoretical approach to implement the
shape and size of the initial geometrical imperfections. However, conventional methods assume that
imperfections should be obtained through the form of classical linear eigen modes of the perfect
structures. This research considers the effects of crookedness and global initial deflection in brace
member on behavior of the structure. A computer program developed to generate crookedness as
well as initial deflection along the brace member. Applied overall initial deflection is plotted in Fig.
4(a). It was approximated in shape of circular arc along the longitudinal axis of the brace member.
Imperfection amplitude of L/1000 (L is length of the brace member) was prescribed in middle of the
brace. As shown in Fig. 4(b), Initial crookedness in flange and web plates applied in form of
sinusoidal wave. As seen for I-beam section, initial local imperfection amplitude is B/200 for flange
plate, and B/150 for web plate.
(a) Initial deflection (b) Initial crookedness of component plates
Figure 4: Initial imperfections in brace member
L
L/1000

118
III. ANALYSIS RESULTS
Nonlinear time history analysis conducted to investigate seismic structural response of
previously mentioned 9 specimens using ABAQUS software package. Von Mises yield criteria with
isoperimetric hardening is employed in this study. Two seismic waves provided by JSHB [9] at
“Level II” caused by inland faults in region with ground “Type I” [N-S and E-W components of
Kobe earthquake (1995)] were input in dynamic response analysis. Ground acceleration data of Type
II-I-1 and Type II-I-2 waves which are applied in this paper are shown in Fig. (5). As seen in Fig. (6),
Kobe N-S component wave was applied in-plane direction and Kobe E-W component wave was
applied in out-of-plane direction. Dynamic analysis performed for both shell and beam model types.
Results are obtained and plotted to compare between various structural models.
(a) Type II-I-1 (Kobe N-S) (b) Type II-I-2 (Kobe E-W)
Figure 5: Input seismic waves
(a) In-plane seismic wave (b) Out-of-plane seismic wave
Figure 6: Seismic wave conditions
Displacement response
Type II-I-1 (Kobe N-S)
Displacement response
Type II-I-2 (Kobe E-W)

119
3.1 Eigen value analysis
Before conducting time history analysis, structural dynamic characteristics should be
determined to compute essential parameters used for dynamic analysis. In order to show natural
mode shapes and frequencies of structures with each specimen, eigenvalue analysis was performed.
Lumped masses equal to super-structural load were considered in corners of the rigid frame.
Analysis results of the specimen 100-6-6 up to 10th mode shape for both shell and beam model are
shown in Table 2 and Table 3, respectively. According to effective mass ratio and mode shapes
illustrated in Fig. 7, it was found that for this case study, in both model types the structure naturally
tends to vibrate at mode shapes with lower frequencies in out-of-plane and in-plane directions. These
two mode shapes were prominent for all cases as well. Comparison of modal frequencies between
two types of shell model and beam one is illustrated in Fig. 8. It revealed good correlation in first
three lowest frequencies. In shell model some frequencies were higher than corresponding values in
beam model. However, these modes do not play a major role in vibration of the structure under
dynamic load conditions. As seen prominent frequencies (mode 1 and mode 3) were almost the
same in both model types.
(a) Mode1(prominent) (b) Mode 2 (c) Mode 3(prominent) (d) Mode 4
Figure 7: Mode shapes of the model made of shell elements (Specimen 100-6-6)
Table 2: Eigenvalue analysis results
(Shell model)
Table 3: Eigenvalue analysis results
(beam model)
Mode
No.
Frequency Period Effective mass ratio (%)
(Hz) (s) X Y Z
1 3.383 0.296 0 0 100
2 5.168 0.193 0 0 0
3 8.712 0.115 100 0 0
4 32.059 0.031 0 0 0
5 32.094 0.031 0 0 0
6 41.555 0.024 0 0 0
7 41.644 0.024 0 100 0
8 45.940 0.022 0 0 0
9 45.942 0.022 0 0 0
10 58.183 0.017 0 0 0
Mode
No.
Frequency Period Effective mass ratio (%)
(Hz) (s) X Y Z
1 3.324 0.301 0 0 100
2 5.108 0.196 0 0 0
3 8.635 0.116 100 0 0
4 31.368 0.032 0 0 0
5 31.368 0.032 0 0 0
6 33.446 0.030 0 0 0
7 33.456 0.030 0 0 0
8 41.405 0.024 0 0 0
9 41.487 0.024 0 100 0
10 56.753 0.018 0 0 0

120
3.2 Time history response for in-plane seismic wave
Above described analytical models were subjected to Type II-I-1 (Kobe N-S component-
1995) in in-plane direction as it shown in Fig. 6(a). Structural damping was considered based on
commonly used Rayleigh damping method. The damping matrix C is assumed to be proportional to
the Mass M and stiffness K matrices, as C=α.M+β.K. α and β factors were calculated using the
following formula.
fi and fj denotes major modal frequencies. hi and hj are damping coefficients of prominent
modes. In order to compare structural responses and observing buckling effects on seismic behavior
of the models, displacement response of the top point as shown in Fig. 6 and base shear force
response are plotted in Fig. 9 to Fig. 11. As mentioned in Table 1 brace member’s slenderness ratio
and width-to-thickness ratio of cross section plate components varied between different specimens.
Fig. 9 shows time history responses for brace member with slenderness ratio of L/r=100. Fig. 10
plots the results for the case of brace member with L/r=90, and Fig. 11 indicates seismic responses
for L/r=80.
( )
( )22
4
ji
jiijji
ff
fhfhff
−
−
=
π
α (4)
( )22
ji
jjii
ff
fhfh
−
−
=
π
β (5)
(a) Specimen 100-6-6 (b) Specimen 100-8-8 (c) Specimen 100-10-10
(d) Specimen 90-6-6 (e) Specimen 90-8-8 (f) Specimen 90-10-10
(g) Specimen 80-6-6 (h) Specimen 80-8-8 (i) Specimen 80-10-10
Figure 8: Comparison of frequency results between shell model and beam model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8 9 10
Frequency(Hz)
Order of Mode
Beam model
Shell model

121
(a) Displacement response (100-6-6) (b) Base shear response (100-6-6)
(c) Displacement response (100-8-8) (d) Base shear response (100-8-8)
(e) Displacement response (100-10-10) (f) Base shear response (100-10-10)
Figure 9: Seismic response to Kobe N-S component earthquake record, R=0.80
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 5 10 15 20 25 30
DisplacementResponse(m)
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model

122
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model

123
According to Fig.9 to Fig.11 history responses for the same seismic wave were different
between shell model and beam model. As illustrated in Figs. 9-11 (e,f) peak responses belonged to
shell model. Fig.12 also shows history of base force versus displacement for in-plane base excitation.
As clearly seen in Figs. 12(c,f,i) when Rf=1.0 and Rw=1.0 larger maximum displacement responses
were observed in case of shell model . It was found that regardless of slenderness ratio of brace
member, beam model is not reliable in case of applying brace member made of component plate with
high width-to-thickness ratio. For specimen 80-6-6, almost no instabilities were found in history
responses.
(a) 100-6-6 (b) 100-8-8 (c) 100-10-10
(d) 90-6-6 (e) 90-8-8 (f) 90-10-10
(g) 80-6-6 (h) 80-8-8 (i) 80-10-10
Figure 12: Time history base shear versus displacement [in-plane seismic wave (Kobe N-S)]
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.01 -0.005 0 0.005 0.01
Baseshear(x106)
Displacement (m)
Beam model
Shell model
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model

124
3.3 Time history response for out-of-plane seismic wave
Type II-I-2 (Kobe E-W component) in out-of-plane direction as it shown in Fig. 6(b) was
used to perform nonlinear time history analysis. Since the ground motion wave direction is
perpendicular to the rigid frame plane, using various brace members had less effect on structural
response. Results for specimen 100-6-6 are plotted in Fig. 13. Plastic residual displacement was
observed due to nonlinear instabilities. Significant buckling effects in base columns caused sudden
drop of displacement response as shown in Fig. 13 (a). Opposite to in-plane seismic wave, as seen in
Fig. 13(b) base shear response did not decreased severely during the second half of the history.
3.4 Local deformation caused by out-of-plane seismic wave
Severe local buckling effects in base column caused difference in vertical surface strain in
shell elements. Significant local deflection confirmed by obtaining vertical stress-strain curve in
outside and inside face of the elements El:1 and El:2 in base column as shown in Fig. 15.
Figure 14: Local deflection in base column for models subjected to out-of-plane seismic wave
In Fig. 15 σY and σV denote yield stress and vertical stress respectively. εV presents vertical
strain and the term εy points out to yield strain. The position of El:1 and El:2 are illustrated in Fig. 14.
Based on output results not any residual strain occurred in case of in-plane excitation using ‘Kobe E-
El:1 El:2
Figure 13: Seismic response to Kobe E-W component earthquake record, R=0.80
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 5 10 15 20 25 30
Time (s)
Beam model
Shell model
-0.4
-0.2
0
0.2
0.4
0 5 10 15 20 25 30
Baseshear(x106N)
Time (s)
Beam model
Shell model

125
W component earthquake record. As seen compression strain observed in outside surface of the
element El:1 and tension strain developed in outside face of the element El:2. In both elements the
compression strain is larger than the tension strain. It is because of existence of vertical loads
assumed to represent super structural weight. In El:2 the strain grew steadily. However in El:1 more
loops were observed since it directly affected by internal compressive forces generated by ground
motion. Based on output results not any residual strain occurred in case of in-plane excitation using
‘Kobe E-W component earthquake record.
IV. CONCLUSION
Model of steel rigid frame with converted V shape brace member with various slenderness
ratio and different width-to-thickness ratio studied to investigate the effect of local deflections on
history response of the structure. In order to obtain better understanding of local buckling effects
shell model adopted as well as beam model. Two results were compared to draw following
conclusions.
1- Regardless of slenderness ratio, larger maximum displacement responses were observed in
case of shell model with higher width-to-thickness ratios Rf=1.00, Rw=1.00.
2- No instabilities observed in history responses for model with lowest slenderness ratio L/r=80
and lowest buckling parameter Rf=0.60, Rw=0.60. However, effects of local deflections caused
instabilities for models with slenderness ratio of larger than L/r=80.
3- Since the brace members are not very effective in perpendicular stiffness of the structure,
sever buckling deformation accrued as the model subjected to out-of-plane ground motions which
lead in residual plastic displacement response.
4- Buckling effects may be confirmed through outside and inside surface strain. In this study
maximum case amounted to 30 time larger than yield strain.
(a) Hysteresis stress-strain in El:1 (b) Hysteresis stress-strain in El:2
Figure 15: Local deflections in base column, R=0.80, 100-6-6
In case of applying out-of-plane seismic wave of Kobe E-W component
-2
-1
0
1
2
-40 -30 -20 -10 0 10 20
σσσσv/σσσσy
εεεεv/εεεεy
Outside Inside
-2
-1
0
1
2
-40 -30 -20 -10 0 10 20
σσσσv/σσσσy
εεεεv/εεεεy
Inside
Outside

126
REFERENCES
[1] T. Yamao, S. Takaji. and S Atavit, “3 Dimensional Seismic Behavior of Deck Type Arch
Bridges with Curved Pair Ribs”, Proceedings of 8th International Conference on STEEL,
SPACE & COMPOSITE STRUCTURES, 2006, K. Lumpur, Malaysia, pp.213-220.
[2] T. Yamao, S. Atavit and Chandra F, “Seismic Behavior and a performance Evaluation of
Deck-Type Steel Arch Bridges under the Strong Earthquakes”, 5th International Symposium
on Steel Structures ISSS’09, 2009, Seoul, Korea.
[3] O. Mohamed, Y Nakamura, T. Yamao and T. Sakimoto, “Performance Evaluation Method
for a Seismic Design of Deck-type Steel Arch Bridges”, Journal of Construction Steel, 2006,
Vol.14, pp.83-90.
[4] B. W. Schafer, Local, Distortional, and Euler Buckling of Thin-Walled Columns, Journal of
Structural Engineering/ March 2002. p289-299.
[5] Tetsuya Yabuki, Janice J. Chambers, Yasunori Arizumi, Tetsuhiro Shimozato, Hiroaki
Matsushita, Buckling capacity of welded stainless steel flanges by finite element analysis,
Engineering Structures 49 (2013) 831–839.
[6] B. W. Schafer, Z. Li, C.D. Moen, Computational modeling of cold form steel, Thin-walled
structures 48 (2010) 752-762.
[7] O. Mohamed, T. Sakimoto, T. Yamao, Ductility of stiffened steel box member, Advanced
steel structure, 2005.
[8] T. Yamao, A. Sujaritpong, Yoshie Tsujino, Evalustion of ultimate strength and strain of I-
section steel members, The 4th International conference on advances in structural engineering
and mechanics (ASEM’08), Jeju, Korea, May 26-28, 2008.
[9] Japan Road Association, “Specifications for Highway Bridges”, Part I - Steel Bridge and Part
V - Seismic Design, Japan, 2002.
[10] Dassault Systèmes Simulia Corp., “ABAQUS 6.11, Abaqus/CAE User’s Manual”.
Providence, RI, USA, 2011.

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