VERIFICATION OF THE METHOD FOR IMPROVING …€¦ · · 2016-01-05Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe 48 [email protected] has reached the ultimate state because it does

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 6, Issue 10, Oct 2015, pp. 46-61 Article ID: IJCIET_06_10_005

Available online at

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=6&IType=10

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication

VERIFICATION OF THE METHOD FOR

IMPROVING ACCURACY OF SIMPLIFIED

SEISMIC RESPONSE ANALYSIS OF STEEL

RIGID FRAME VIADUCTS

Tatsuo Kakiuchi

JR West Japan Consultants Company (PhD Candidate),

5-4-20 Nishinakajima, Yodogawa-Ku, Osaka, 532-0011, Japan

Akira Kasai

Associate Profesor, Kumamoto University, 2-39-1 Kurokami,

Chuo-Ku, Kumamoto, 860-8555, Japan,

Shohei Okabe

GSST, Kumamoto University, 2-39-1 Kurokami, Chuo-Ku,

Kumamoto, 860-8555, Japan,

ABSTRACT

This study is aimed at verifying the usefulness of the estimation method

which is able to permit several plastic hinges' occurring to the steel rigid

frame viaduct which is higher statically indeterminate. For this purpose, the

analytical model, named “whole system model”, which applies shell elements

to the place where plastic hinges may form, when the structures are subjected

to severe earthquakes, was constructed. Next, Pushover analysis using the

whole system model was carried out for evaluating seismic performance of

this structure, and simplified seismic response analysis using an equivalent

single-degree-of-freedom system model and skeleton curve which resembles a

bilinear model based on a result of the Pushover analysis was performed. And

then, the results of simplified earthquake response analysis was compared

with the result of dynamic analysis using whole system model. Finally, it was

contrived to apply a trilinear model to skeleton curve on restoring force-

horizontal displacement relationship as a way to improve the predictability of

the response displacement of this structure. This paper also examines how the

accuracy of seismic response analysis using the equivalent single-degree-of-

freedom system model could be improved.

Key words: Combined static-dynamic numerical method for seismic response,

Plastic hinge, Pushover analysis, Seismic performance evaluation, Steel rigid

frame viaducts

Verification of The Method For Improving Accuracy of Simplified Seismic Response

Analysis of Steel Rigid Frame Viaducts


Cite this Article: Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe.

Verification of The Method For Improving Accuracy of Simplified Seismic

Response Analysis of Steel Rigid Frame Viaducts. International Journal of

Civil Engineering and Technology, 6(10), 2015, pp. 46-61.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=6&IType=10

1. INTRODUCTION

Steel structures built in Japan have been required to have excellent seismic

performance that can resist the Hyogo Earthquake in 1995, the Tohoku-Pacific Ocean

Earthquake in 2011 or a major earthquake like the Tokai, Tonankai, Nankai

Consolidated Type Earthquake which is predicted to occur in the near future. In the

structures such as these, the steel rigid frame viaducts rigid-connected between

superstructure and piers which is focused on this paper, is one of the structures which

can improve seismic performance. This structure has the following characteristics; 1)

the height of the viaduct part can be lowered, 2) flexible correspondence is easy for

the vertical linear shape of the railroad, 3) support system between box girder and

piers can be omitted, and so on. Recently, a viaduct having these advantages is

adopted in a steel bridge for railway in Japan. The schematic view of the viaduct to

intend for in this study is shown in Fig. 1.

The bridge type in this study has the longitudinal direction of the bridge and the

out-of-plane direction of the piers in the same orientation. Beams and corners of the

piers of such viaducts are exposed to torsion in addition to bending during an

earthquake due to horizontal inertia. For an investigation of the members subjected to

complex loading, Kasai et al. [1] examined basic data from seismic safety evaluation

on the beams and corners of rigid frame viaduct piers where shear would prevail.

Nakai et al. [2], [3], [4] evaluated both experimentally and analytically seismic

capacity of a box section subjected to torsion and bending simultaneously. However,

these studies were focused on evaluating individual segments.

Conventional seismic performance evaluation is based on the assumption that the

ultimate state of the whole structure is reached when at least the ultimate state of a

single segment. However, this bridge which is a higher statically indeterminate

structure can remain sufficiently safe in terms of seismicity when a single segment

Longitudinal

Vertical

Transverse

Figure 1 Conceptual diagram of bridge system in this study

Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe


has reached the ultimate state because it does not lead to reduction in the seismic

capacity of the whole structure as long as the member components have adequate

plate thickness and stiffness. Seismic performance of the structure therefore can be

underestimated by the conventional segment-based evaluation. Focusing on this point,

the authors [5] proposed a seismic performance evaluation method using shell

elements. The conventional displacement-based verification method [6] is used for a

conventional equivalent single-degree-of-freedom system structure. Then, the

proposed method [5] is the displacement-based verification method for a higher

statically indeterminate structure. Pushover analysis using the whole system model to

consider local buckling behavior was carried out, and simplified seismic response

analysis using an equivalent single-degree-of-freedom system model and skeleton

curve which resembles a bilinear model based on a result of the Pushover analysis

was performed. In this combined static-dynamic numerical method, the most

important characteristic of the proposed method is the use of shell elements for

Pushover analysis which considers local buckling behavior.

It is effective to handle a higher statically indeterminate structure as a whole

system as proposed, not by segment as conventionally done, in seismic performance

evaluation. The proposed method is carried out seismic response analysis using the

equivalent SDOF model to obtain response values. However, the applicability of the

equivalent SDOF model has not yet been well studied. The purpose of this study is to

verify the usefulness of the seismic performance evaluation method developed for a

higher statically indeterminate structure to allow for plastic hinges being set at

multiple locations.

2. DESIGN CONDITIONS AND ANALYSIS MODEL

Table 1 shows the design conditions of the bridge. The bridge is a four-span steel

rigid frame viaduct having the pier-superstructure integrated system as shown in Fig.

1. The details are described in reference [5]. Fig. 2 shows the general view of the

analysis model. The segments modeled with shell elements are the base of a viaduct

pier, the beam of a viaduct pier near the rigid connection with the superstructure and a

corner closest to the superstructure. The shell elements for the base and the beam

were input with initial deflection. The initial deflection waveforms were created by

expressing each of general and stiffener local initial deflections as sine waves of the

first mode and overlapping them together. The maximum initial deflection was

determined based on the guidelines for buckling design [7]. As the material

constitutive rule of steel, multilinear strain hardening rule having a yield plateau was

used in the Pushover analysis, and the bilinear skeleton model was used in the seismic

response analysis. A universal finite element analysis software ABAQUS [8] was

used for the analysis.

Table 1 Design conditions

Structural type Steel rigid frame viaduct

Span length 183.4 m

Spans 39.2 m + 40.0 m + 52.5 m + 51.7 m

Girder width 11.8 m

Girder height 2.8 m




Figure 2 Analysis model

3. COMBINED STATIC-DYNAMIC NUMERICAL METHOD

3.1 Pushover analysis

Pushover analysis and seismic response analysis using the equivalent SDOF model as

shown in Fig. 3 were carried out following the method proposed in reference [5] for

seismic performance evaluation of the bridge. Fig. 4 shows the load-displacement

relationship obtained by the Pushover analysis, where horizontal load H is the sum of

reaction forces at all pier bases, and horizontal displacement δ is displacement at the

center of gravity of the superstructure. Displacement at the maximum load is used as

ultimate displacement δu, with the ultimate state assumed to be reached under the

maximum load. In higher statically indeterminate structures like the bridge, local

buckling should have occurred at many locations when the capacity of the structure

starts to decrease. Evaluation to the maximum load state in this study was an attempt

to ensure adequate safety margin in the design.

Figure 3 Equivalent SDOF model Figure 4 Load-displacement curve



Figure 5 Hysteresis Figure 6 Response spectra of earthquake (h=0.05)

3.2 Seismic response analysis using the equivalent SDOF model

The Restoring force characteristic to be used for the equivalent SDOF model were

obtained from the results of the Pushover analysis using the whole system model [9].

When restoring force characteristics has been made approximately, it decided to meet

the following two conditions certainly. Those are, 1) that both of bilinear approximate

model and whole system model are same about initial horizontal stiffness which

indicates the relationship between restoring force in horizontal direction and the

horizontal displacement, and 2) that the restoring force and the displacement which

indicate the ultimate state in whole system model are also passed certainly in the

bilinear approximate model. The first break point was determined in accordance with

reference [6] so that energy absorption amount would be constant until the ultimate

point. Restoring force characteristic was applied to the hysteresis of the equivalent

SDOF model as shown in Fig. 5.

Fig. 4 shows the load-displacement relationship in the whole system model and

the restoring force characteristic used for the equivalent SDOF model. Table 2 shows

the parameters of the restoring force characteristic, where K = elastic stiffness, K =

elasto-plastic stiffness, Hy = yield horizontal load, δy = yield displacement, Hu =

ultimate horizontal load and δu = ultimate displacement.

3.3 Time history of response displacement

The analysis used epicentral ground motion specified as Level-2 earthquake in the

Design Standards for Railway Structures and Commentary (Seismic Design) [10].

Analysis was performed also for the El Centro (NS) and Taft (EW) waves. Fig. 6

shows response spectra of earthquakes. The dumping ratio h = 0.05 were used in the

present analysis. Figs. 7 show the seismic motion. Figs. 8 show the time history of

response displacement at the top of the Pier P3 in the equivalent SDOF models.

Table 3 shows maximum response displacement δmax and its ratio to ultimate

displacement, δmax/δu. According to the seismic response analysis using the equivalent

Table 2 Parameters of the restoring force characteristic

K [kN/mm] ζK [kN/mm] Hy [kN] δy [mm] Hu [kN] δu [mm]

706 24.6 7.01×104 99.2 8.58×10

4 740




SDOF model, maximum response displacement δmax was 270 mm for Level-2

earthquake, 90 mm for the El Centro (NS) wave, and 46 mm for the Taft (EW) wave.

Figure 7 Seismic motion Figure 8 Time history of response

displacement

Table 3 Maximum response displacement

δmax [mm] δmax /δu

Equivalent SDOF model

Level-2 earthquake 270 0.365

El Centro (NS) 90 0.115

Taft (EW) 46 0.062



4. ELASTO-PLASTIC SEISMIC RESPONSE ANALYSIS USING

THE SHELL-ELEMENT WHOLE SYSTEM MODEL

4.1 Seismic response analysis using the whole system model

This section describes the results of the seismic response analysis using the whole

system model and compares them with the results of the combined static-dynamic

numerical method described in the previous section. Morishita et al. [11] inspected the

precision of the whole system model using the shell element in a simple steel pier,

according to the seismic response analysis. Figs. 9 show the time history of response

displacement at the top of the Pier P3 in this structure. The response displacement was

determined as relative displacement between the superstructure and the base. Figs. 10

shows the load-displacement history, where load H is the base shear. The results with

the equivalent SDOF model are also shown for comparison. Table 4 shows the values

of maximum response displacement δmax.

4.2 Comparison of the time history of response displacement

Maximum response displacement δmax for Level-2 earthquake was 226 mm in the

whole system model and 270 mm in the equivalent SDOF model, being about 19%

larger in the equivalent SDOF model. The response displacement history showed that

the results of the two models fitted well until around 2 seconds and started to differ at

past 2 seconds. The amplitude was found to be larger in the whole system model

during the period from 2 to 6 seconds and then larger in the equivalent SDOF model

during the period from 6 to 10 seconds. Center of the amplitude range in the whole

system model moved significantly in the positive direction in about 4 seconds, while

that in the equivalent SDOF model during the same period showed no significant

movement because of the small amplitude. As a result, center of the amplitude in the

equivalent SDOF model stayed in the negative side, allowing for an interpretation that

maximum response displacement was larger than that in the whole system model.

Residual displacement also showed a tendency of being larger in the equivalent SDOF

model.

Maximum response displacement δmax for the El Centro (NS) wave was 85 mm in

the whole system model and 90 mm in the equivalent SDOF model, being about 6%

larger in the equivalent SDOF model. The response displacement history showed that

the results of the two models fitted well until around 2 seconds and started to differ at

past 2.5 seconds. Maximum response displacement δmax for the Taft (EW) wave was

46 mm in the whole system model and 46 mm in the equivalent SDOF model,

showing no differences between the two models. The response displacement history

showed that the results of the two models fitted well until around 5 seconds.

Consequently, the maximum response displacement of the equivalent SDOF

model that the restoring force characteristic was made the bilinear skeleton model is

bigger than the maximum response displacement of the whole system model. Therefore, the seismic performance of the steel rigid frame viaduct which is higher

statically indeterminate can be judged to compare the ultimate displacement δu with

the maximum response displacement δmax of the equivalent SDOF model that the

restoring force characteristic was made the bilinear skeleton model.




Figure 9 Time history of response displacement Figure 10 Load-displacement history


δmax [mm]

Whole system model Equivalent SDOF model

Level-2 earthquake 226 270 [119%]

El Centro (NS) 85 90 [106%]

Taft (EW) 46 46 [100%]



4.3 Comparison of the load-displacement history

In Figs. 10, the whole system and equivalent SDOF models were found to show well

fitting results in seismic response analysis until yield displacement. In case of Level-2

earthquake, the loops of different patterns appeared in the load-displacement curves of

the two models from the point where the whole system model yielded. Load was

found to be larger in the equivalent SDOF model, with the center of vibration shifted

away from the origin under plastic deformation. This was likely due to some factor in

the restoring force characteristic used for the equivalent SDOF model. The restoring

force characteristic had been obtained by the bilinear model to skeleton curve on

restoring force-horizontal displacement relationship based on a result of the Pushover

analysis as described in Section 3. However, stiffness reduction after the yield

displacement was slow in the Pushover analysis values. Rough approximation of such

a curve with a bilinear model will result in a significant discrepancy between the

analytical values and approximate values, greatly affecting the seismic response

analysis results.

In case of the El Centro (NS) wave, the difference of the maximum response

displacement δmax between the both models has occurred a little, because the

maximum response displacement δmax becomes near yield-displacement. In case of the

Taft (EW) wave, the maximum response displacement δmax between the both models

is almost similar, because the maximum response displacement is almost half of the

yield-displacement.

Consequently, both models were found to show exhibit different behaviors after

yielding by using the bilinear skeleton model. To improve the accuracy of seismic

response analysis using the equivalent SDOF model, for Level-2 earthquake and the

El Centro (NS), the authors modified the restoring force characteristic into a trilinear

model as described in the following section.

5. EXAMINATION TO IMPROVE ACCURACY OF THE

EQUIVALENT SDOF MODEL

5.1 Skeleton of the equivalent SDOF model modified by the trilinear

skeleton model

In this chapter, case study for improving the accuracy of estimation for maximum

response displacement in the typical location of this structure is discussed. The time

history of response displacement for the whole system and equivalent SDOF models

is compared in the previous section. And the load-displacement relationship is

compared. As the result, there is a possibility that the accuracy of the maximum

response displacement δmax isn't good because both the behavior isn't fitting well. This

was likely due to the influence of the restoring force characteristic used for the

equivalent SDOF model. The authors modified the restoring force characteristic as

described in this section in an attempt to improve the accuracy of seismic response

analysis. More accurate seismic response analysis would be obtained by using more

accurate approximates of the Pushover analysis results for the restoring force

characteristic.

Accordingly, the flow chart shown in Fig. 11 was developed at first. The outline

of this flow chart is as follows.




1) Restoring force-horizontal displacement relationship is expressed by bilinear

model approximately using the Pushover analysis of this structure according to

Kakiuchi et al. [5].

2) Seismic response analysis is carried out using an equivalent SDOF model and

above relationship.

3) Restoring force-horizontal displacement relationship is upgraded using trilinear

skeleton model, if accuracy of this numerical result is required.

4) The second break point in the modified model was set at the point of 1.1×δmax, B

so that a safety margin of 10% was added to maximum response displacement

δmax, B. Initial stiffness and the second point were fixed in the determination of

restoring force characteristic. The first break point was set so that energy

absorption amount would be constant until the second break point. As in the

model before the modification, the same restoring force characteristic was applied

to the hysteresis.

Figure 11 Flowchart for improvement of the equivalent SDOF model

Yes

No

Redesign of sections

Yes

No

Pushover analysis

( K, Hu, u )

Start

Approximation for hysteresis

(K, Hy, y )

n = 1 Bilinear (Reference [5])

Dynamic analysis by E-SDOF model

(max )

End

Structural properties and materials

Requirement of accurate max

Verification of displacement

max ≤ u

Modified approximation for hysteresis

( K, Hy, y )

n ≥ 2 Trilinear (This study)



Figs. 12 and Table 5 shows the restoring force characteristic into a trilinear

skeleton model for Level-2 earthquake and the El Centro (NS) wave. And the value of

the 1st brake point and the 2nd brake point is indicated.

Table 5 Brake points of trilinear skeleton model

Level-2 earthquake EL Centro

1st brake

point

2nd brake

point

Ultimate

state

1st brake

point

2nd brake

point

Ultimate

state

Bilinear [mm]

H [kN]

99.2

7.01×104

- 740

8.58×104

99.2

7.01×104

- 740

8.58×104

Trilinear-1 [mm]

H [kN]

75.7

5.35×104

300

7.68×104

740

8.58×104

56.2

3.94×104

100

5.30×104

740

8.58×104

Trilinear-2 [mm]

H [kN]

74.0

5.23×104

270

7.46×104

740

8.58×104

58.3

4.12×104

110

55.2×104

740

8.58×104

5.2 Modification of the equivalent SDOF model

Figs. 13 show the time history of response displacement in the whole system and

equivalent SDOF models. Figs. 14 show the load-displacement history. Table 6 shows

maximum response displacement δmax of each model. The values in square brackets in

the table are percentages which are the ratios of maximum response displacement of

the equivalent SDOF model to that of the whole system model. The equivalent SDOF

model with the modified the restoring force characteristic using trilinear skeleton

model was found to exhibit behavior closer to that of the whole system model.

Maximum response displacement δmax for Level-2 earthquake was improved to an

error rate of 7%, with a good fitting found in the load-displacement history between

the two models. Maximum response displacement δmax for the El Centro (NS) wave

was also improved to an error rate of 11%, again with a good fitting found in the load-

displacement history between the two models.

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]

Pushover analysis Bilinear Ultimate state Trilinear-1 2nd break point Trilinear-2 2nd break point

(a) Level-2 earthquake (b) El Centro (NS)

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]


Figure 12 Trilinear skeleton model




Figure 13 Time history of response t Figure 14 Load-

Displacemen Displacement history


δmax [mm]

Whole system

model


Bilinear

(n=1)

Trilinear-1

(n=2)

Trilinear-2

(n=3)

Level-2 226 270 [119%] 242 [107%] 243 [108%]

El Centro (NS) 85 90 [106%] 96 [113%] 94 [111%]

Moreover, ground motions which the acceleration of the design earthquake

motion is set to 0.75, 1.5, 1.75 and 2.0 times uniformly are prepared, to inspect the

effect of this simple method to an earthquake motion with the size of the various

accelerations. Although some results when these earthquake motions are subjected to

it, are gathered by an appendix specifically, results about the maximum response

displacements are summarized in Table 7. The accuracy of the maximum response

displacement δmax can obtain by iterating total of 3 times of the once by the bilinear

skeleton model and twice by the trilinear skeleton model.



Table 7 Approximation of the restoring force characteristic

Whole system

mode


Bilinear

(n=1)

Trilinear-1

(n=2)

Trilinear-2

(n=3)

max,W [mm]

(1)

max,W

/ u

max,B [mm]

(2)

max,B

/ u

(2)-(1)

max,T-1 [mm]

max, T-1

/ u

max,T-2

[mm]

(3)

max, T-2

/ u

(3)-(1)

El Centro 85 0.015 90 0.122 0.007 96 0.130 94 0.127 0.012

L2×0.75 - - 153 0.207 - 166 0.224 167 0.225 -

L2 226 0.305 270 0.365 0.060 242 0.327 243 0.328 0.023

L2×1.5 439 0.593 454 0.614 0.021 403 0.545 407 0.550 -0.043

L2×1.75 - - 537 0.726 - 533 0.720 533 0.720 -

L2×2.0 - - 674 0.911 - 674 0.911 674 0.911 -

u = 740 mm : The ultimate displacement

max,W : The maximum response displacement of whole system model

max, B : The maximum response displacement of E-SDOF model by the bilinear model (n=1)

max, T-1 : The maximum response displacement of E-SDOF model by the trilinear model

(n=2)

max,T-2 : The maximum response displacement of E-SDOF model by the trilinear model (n=3)

Fig. 15 shows the each maximum response displacement δmax for Level-2

earthquake, 1.5 times of Level-2 earthquake and the El Centro (NS) wave in the case

of each model. In Table 5 and Figs. 12, it was found that the prediction method which

bilinear skeleton model was used as a restoring force-displacement relationship of the

viaduct dealt with this study is very close to seismic response displacement max, W of

this structure using whole system model. Whichever earthquake motions treated by

this study was used, it was found that the maximum response displacement obtained

using the equivalent SDOF model was larger than the case of the whole system

model, when the displacement of this structure predicted using a bilinear model as

restoring force-displacement relationship. The gap of ratio of δmax/δu was within

about 6% of the ultimate displacement between the prediction displacement by

bilinear approximation model and the prediction displacement by the whole system.

Figure 15 Maximum response displacements-maximum earthquake motions relationship




It was found that the accuracy of the response displacement improved in most

cases, when the displacement predicted using a triilinear model as restoring force-

displacement relationship. In this case, the gap of ratio of δmax/δu was within about

4% of the ultimate displacement between the prediction displacement by trilinear

approximation model and the prediction displacement by the whole system model as

shown in Table 7.

Therefore, the equivalent SDOF model using the restoring force characteristic

approximated by the bilinear skeleton model is an effective method in order to verify

the seismic performance of the higher statically indeterminate structure like this steel

rigid frame viaduct.

6. CONCLUSION

This study is aimed at verifying the usefulness of a seismic performance evaluation

method developed for a higher statically indeterminate structure to allow for plastic

hinges being set at multiple locations. Pushover analysis using the whole system

model was carried out for evaluating seismic performance of this structure, and

simplified seismic response analysis using an equivalent single-degree-of-freedom

system model. The major findings were as follows:

(1) The flow which applies a result of the Pushover analysis to restoring force-

displacement relationship was proposed to carry out seismic response analysis using

the equivalent SDOF model for the high-level indeterminate structures.

(2) To improve the precision of the response displacement which predicted restoring

force-displacement relationship by a response prediction model that resembles a

bilinear model more, the method which resembles a trilinear model was developed as

shown in Fig. 11.

(3) It was found that the prediction method which bilinear skeleton model was used as a

restoring force-displacement relationship of the viaduct dealt with this study was very

close to seismic response displacement of this structure using whole system model.

(4) Whichever earthquake motions treated by this study was used, it was found that the

maximum response displacement obtained using the equivalent SDOF model was

larger than the case of the whole system model, when the displacement of this

structure predicted using a bilinear model as restoring force-displacement

relationship. The gap of ratio of δmax/δu was within about 6% of the ultimate

displacement between the prediction displacement by bilinear approximation model

and the prediction displacement by the whole system.

(5) The gap of ratio of δmax/δu was within about 4% of the ultimate displacement

between the prediction displacement by trilinear approximation model and the

prediction displacement by the whole system model.

(6) The equivalent SDOF model using the restoring force characteristic

approximated by the bilinear skeleton model is an effective method in order to

verify the seismic performance of the higher statically indeterminate structure

like this steel rigid frame viaduct.

(7) The equivalent SDOF model using the bilinear skeleton model is an effective

technique for practical design that redesign cross sections iteratively on

verifying cross sectional force of members.



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[10] Railway Technical Research Institute, Design Standards for Railway

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[11] K. MORISHITA, T. USAMI, T. BANNO, A. KASAI, Applicability on Dynamic

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Technology, 6(1), 2015, pp. 113 - 126.




APPENDIX

In this appendix, ground motions which the acceleration of the design earthquake

motion is set to 0.75, 1.5, 1.75 and 2.0 times uniformly are prepared, to inspect the

effect of this simple method to an earthquake motion with the size of the various

accelerations.

(d) L2×2.0 (d) L2×2.0 (d) L2×2.0

0 5 10 15-800

-600

-400

-200

0

200

400

[

mm

]

Time [sec]

E-SDOF model [Bilinear] E-SDOF model [Trilinear-1,2]

-500 0 500-100000

-50000

0

50000

100000

H [

kN

]

[mm]

E-SDOF model [Bilinear]

E-SDOF model [Trilinear-1,2]

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]

Pushover analysis Bilinear Ultimate state Trilinear-1,2 2nd break point

(a) L2×0.75 (a) L2×0.75 (a) L2×0.75

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]


0 5 10 15

-200

-100

0

100

200

[

mm

]Time [sec]

E-SDOF model [Bilinear] E-SDOF model [Trilinear-1] E-SDOF model [Trilinear-2]

-300 -200 -100 0 100 200 300-100000

-50000

0

50000

100000

H [

kN

]

[mm]


E-SDOF model [Trilinear-1]


Figs. A2: Time history of

response displacement

Figs. A3: Load-displacement

history

Figs. A1: Load-displacement

curve

(b) L2×1.5 (b) L2×1.5 (b) L2×1.5

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]


0 5 10 15-600

-400

-200

0

200

[

mm

]

Time [sec]

Whole system model E-SDOF model [Bilinear] E-SDOF model [Trilinear-1] E-SDOF model [Trilinear-2]

-500 0 500-100000

-50000

0

50000

100000

H [

kN

] [mm]

Whole system model




(c) L2×1.75 (c) L2×1.75 (c) L2×1.75

0 200 400 600 800 1000

20000

40000

60000

80000

100000

[mm]

H [

kN

]

Pushover analysis Bilinear Ultimate state Trilinear-1,2 2nd break point

0 5 10 15-600

-400

-200

0

200

[

mm

]

Time [sec]

E-SDOF model [Bilinear] E-SDOF model [Trilinear-1,2]

-500 0 500-100000

-50000

0

50000

100000

H [

kN

]

[mm]


E-SDOF model [Trilinear-1,2]

Documents

VERIFICATION OF THE METHOD FOR IMPROVING …€¦ · · 2016-01-05Tatsuo Kakiuchi, Akira Kasai and Shohei Okabe 48 [email protected] has reached the ultimate state because it does