12 seismic behavior of steel rigid frame with …€¦ · keywords: steel rigid frame, seismic...

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 Yamao

2

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

© IAEME: www.iaeme.com/Ijciet.asp

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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

Wt

H

W

B

B

Bf

Hwtw

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/m

m²)

ε

SM400

2

2)1(12

π

νσ

kEt

BR

y

f

f

f

−= (1)

2

2 )1(12

π

νσ

kEt

HR

y

w

w

w

−= (2)

2

2)1(12

π

νσ

kEt

HR

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

−

−=

πα (4)

( )22 jijjii

ff

fhfh

−

−=

πβ (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

Fre

qu

en

cy (

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

Fre

qu

en

cy (

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

Fre

qu

en

cy

(H

z)

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

Fre

qu

en

cy (

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

Fre

qu

en

cy

(H

z)

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

Fre

qu

en

cy

(H

z)

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

Fre

qu

en

cy (

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

Fre

qu

en

cy (

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

Fre

qu

en

cy (

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 30Dis

pla

cem

en

t R

esp

on

se (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Base

sh

ear

(x10

6N

)

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 30Dis

pla

ce

men

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Base s

hear

(x10

6N

)

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 30Dis

pla

cem

en

t R

esp

on

se (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Ba

se s

hea

r (x

10

6N

)

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 30Dis

pla

cem

en

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Ba

se s

he

ar

(x10

6N

)

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 30Dis

pla

ce

men

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Bas

e s

hear

(x1

06

N)

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 30Dis

pla

ce

men

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30B

as

e s

hear

(x1

06

N)

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 30Dis

pla

ce

men

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Bas

e s

hear

(x1

06

N)

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 30Dis

pla

ce

men

t R

es

po

ns

e (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Bas

e s

hear

(x1

06

N)

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

Ba

se s

he

ar

(x10

6)

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

Ba

se s

hea

r (x

10

6)

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

Ba

se

sh

ear

(x10

6)

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

Bas

e s

he

ar

(x1

06)

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

Bas

e s

he

ar

(x1

06)

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

Bas

e s

he

ar

(x1

06)

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

Bas

e s

he

ar

(x1

06)

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

Bas

e s

hea

r (x

10

6)

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

Ba

se

sh

ea

r (x

10

6)

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 30Dis

pla

cem

en

t R

es

po

nse

(m

)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Base s

hea

r (x

10

6N

)

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 30Dis

pla

cem

en

t R

es

po

nse (

m)

Time (s)

Beam model

Shell model

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25 30

Ba

se

sh

ea

r (x

10

6N

)

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

σσ σσv/ σσ σσ

y

εεεεv/εεεεy

Outside Inside

-2

-1

0

1

2

-40 -30 -20 -10 0 10 20

σσ σσv/ σσ σσ

y

εεεεv/εεεεy

Inside

Outside



126

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