seismic design of arch bridges during … design of arch bridges during strong earthquake kiyofumi...

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1 CTI Engineering Co., Ltd., Fukuoka, Japan,Email: k-nakagw @fukuoka.ctie.co.jp2 CTI Engineering Co., Ltd., Fukuoka, Japan,Email: irie @fukuoka.ctie.co.jp3 CTI Engineering Co., Ltd., Fukuoka, Japan, Email: [email protected] Ministry of Construction, Japan

SEISMIC DESIGN OF ARCH BRIDGES DURING STRONG EARTHQUAKE

Kiyofumi NAKAGAWA1, Tatsuo IRIE2, Allan D SUMAYA3 And Kazuya ODA4

SUMMARY

In structural design of arch bridges, it is essential to determine plastic regions generated duringstrong earthquakes and to consider the effect of axial force fluctuation in dynamic responseanalysis.

In this study, nonlinear time history analysis is performed with respect to bridge’s axis andtransversal axis where the bridge is modeled as two-dimensional frame model. Three long periodicand three short periodic acceleration waves of approximately 3.30 m/sec2 and 6.30 m/sec2maximum, respectively, are considered as seismic loads. The natural period in the bridge axis andtransversal axis directions are found to be 1.47 sec and 1.17 sec, respectively, which implies thatlong periodic earthquakes are expected to produce more hazardous response.

Plastic regions generate at the upper end of fixed pier and end posts as well as at the base of archring when seismic force acts in bridge axis. In the transversal axis, plastic regions occur at the baseof pier, end posts and arch ring where its area of generation is wider than in the bridge direction.Also, since natural period in the transversal direction is little short, plastic regions are producedeven in short periodic earthquakes but safety of the bridge is still ensured since the ductility factorobtained is less than three.

Comparing the results of the case where axial force fluctuation is considered with the case where itis neglected, curvature at the base of arch ring increases by 20 percent and sectional force at thebase of pier by 10 percent.

INTRODUCTION

The bridge considered in this study is a concrete deck Langer arch bridge with prestressed stiffening girders. It is300 meters long with 143-meter arch span, which will be constructed over the Takachiho ravine, one of Japan’squasi-national park. Its main conditions considered in design are as follows:1. The bridge must have a span of approximately 140 meters in order to cross Takachiho ravine;2. Aesthetic form of the bridge is important since it will stand on a quasi-national park; and3. Economical value of the bridge.Different types of bridges that satisfy the above conditions were compared and evaluated. Overall evaluationresult shows that reinforced concrete Langer arch bridge with prestressed stiffening girders is the mostappropriate structure.

Seismic design of this bridge for strong earthquakes is performed by nonlinear dynamic analysis. The method ofanalysis and computed results along bridge axis is presented in this paper.

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SUMMARY OF BRIDGE

a) Type of bridge: Reinforced concrete Langer arch bridge with prestressed stiffening girdersb) Bridge length: L=300 metersc) Span: l = 2@40 m + 150 m + 2@35 md) Arch span: 143 meterse) Road width: W=15.5 metersf) Cross section: refer to Fig. 1g) General view of bridge: refer to Fig. 2

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Figure1: Cross section of box girder

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Figure2: Side view of bridge

METHOD OF ANALYSIS

Arch bridges, according to Design Specifications for Highway Bridges-Part V Seismic Design (1996) [1], areconsidered as structures that exhibit complex behaviour during strong earthquakes. Moreover, since it alsostatically indeterminate structures of high degree, nonlinear structural members are expected to generate in morethan one area. This suggests that, in this case, energy constant rule is not valid and ductility design method isinapplicable. Therefore, dynamic response analysis should be carried out, where plastic hinge can be determined.Since the bridge is statically indeterminate, axial force acting on arch ring, vertical members and piers fluctuatedue to horizontal action of earthquake. Effect of axial force fluctuation suggests changing of flexural momentand curvature relationship in the analysis. In this study, the effect of axial force fluctuation is investigated bycomparing response curvature when axial force is maximum and minimum with allowable curvature. Here,relationship of flexural moment and curvature is varied in structural members found to yield when flexuralstrength due to dead load is considered.There are two types of ground motions used in the analysis as indicated in Design Specifications of HighwayBridges-Part V Seismic Design (1996) [1], that is, Type I and Type II. Type I corresponds to plate boundary typelarge-scale earthquakes, which characterized by long periodic waves. On the other hand, Type II corresponds toinland direct strike type earthquake like the 1995 Hyogo-ken Nanbu Earthquake, which characterized by shortperiodic waves. Also, both are ground motions with high intensity, though less probable to occur during theservice period of the bridge. Acceleration spectra for strong grounds (i.e. classified as ground Type I in Design

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Specification of Highway Bridges), acceleration spectra ranges from 2.0 to 7.0 m/sec2 in ground motion Type Iand 0.75 to 200 m/sec2 in Type II.

CONDITIONS FOR ANALYSIS5.

a) Seismic classification of bridge: Class B (i.e. bridges considered with high importance)b) Ground type: Type I (i.e. good diluvial ground and rock mass)c) Regional classification: B class (areas with moderate probability of earthquake occurrence)d) Frame model for dynamic analysis: refer to Fig. 3

Nonlinear beam elements: piers (P1, P2, P3), vertical members (V1, V2), and arch ringLinear beam elements: pier (P4), vertical member (V2 to V5), stiffening girder

Figure 3: Frame model used in the analysis

e) Strength of structural members: refer to Table 1f) Relationship of flexural moment and curvature for nonlinear structural members:

Tri-linear degrading stiffness model (Takeda model) considering strength of reinforced concrete duringcracking, yielding and ultimate stage.

g) Damping constants:Linear structural members: h = 0.05Nonlinear structural members: h = 0.0.2Strain energy types of modal damping constants are calculated using these values. Also, Rayleighdamping is considered for viscous damping in dynamic response analysis.

h) Direct integration:Method: Newmark- β method

Time interval: 0.01/10 = 0.001 sec.Duration considered in the analysis: Total duration of input acceleration + 20 seconds of zeroacceleration

i) Initial sectional force:Sectional forces due to dead load are considered in all structural members except in stiffening girders.

j) Direction of load:Only horizontal ground motion is considered. Also, considering the asymmetrical form of bridge,direction of input acceleration is considered in both sides along bridge axis.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1920

21 22 232425 26 27

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201203204

205

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1009

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2003

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30013003

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600160046005600660076008

6009

601060116012601360146017

301303304305306

307

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312

313314315317

401402404405406408

9001

9003

9004

9005

90069008

9009

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9011

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90199021 9022

90249026

9027

90299030

9031

9032

90339034

90359037

9038

9039

9040

9042

A1

P1

P2

V1

V2

V3V4

V5

V6

P3

P4 A2

R R R

R

H

H

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HH

H

H

R

R

R S

N L

N L

L

L LL

N L

NL

L

L LL

NL

N L

N L

2001

2002

along bridge axis

R : rig id

H : hin ge

S : sli de

L : linear member

NL: non linear member

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Table 1: Strength of structural members

Structural Concrete Steel bars Arrangement of reinforcement bars notesStiffening 40 �/mm2 SD295 �Arch ring 40 �/mm2 SD345 Main bars: D32ctc125(2.0steps),

30 �/mm2 SD345 Main bars: D32ctc125(1.0step), V1, V6

Supe

rstr

uctu

re

Verticalmember 24 �/mm2 SD295 � V2 to V5Pier P1 30 �/mm2 SD345 Main bars: D51ctc150(2.0steps),Pier P2 21 �/mm2 SD295 Main bars: D38ctc125(1.0step),Pier P3 21 �/mm2 SD295 Main bars: D51ctc150(1.0step),

Subs

truc

ture

Pier P4 21 �/mm2 SD295 �6. EIGEN ANALYSIS

Eigen analysis is used to examine the seismic characteristic and viscous damping property of bridge structure.Results are shown in Table 2 and Fig. 4. As indicated in these results, first and eighth vibration modes are foundto be dominant.

Table 2: Eigen Analytic Resultsdegree of

modenatural

frequencynaturalperiod

modaldampingconstant

i (H z) (sec )al ongbr i dgeaxi s

ver t i caldi r ect i on

al ongbr i dgeaxi s

ver t i caldi r ect i on

h i

1 0 .67 8 75 1 .47 3 30 3 6 .9 6 0 1 .68 4 6 1 0 0 .04 5 622 1 .42 7 30 0 .70 0 62 1 .03 1 -1 .0 91 6 1 0 0 .04 9 803 2 .21 7 50 0 .45 0 97 5 .12 9 -0 .9 35 6 2 0 0 .04 9 584 2 .65 9 70 0 .37 5 98 -2 .0 53 1 .19 1 6 2 0 0 .04 9 115 3 .18 2 50 0 .31 4 22 3 .45 5 -11 .27 0 6 3 6 0 .04 9 806 3 .80 0 90 0 .26 3 09 -11 .47 0 -1 .1 60 6 8 6 0 .04 9 627 3 .89 0 90 0 .25 7 01 7 .29 6 1 1 .0 3 0 7 1 1 2 0 .04 9 288 4 .02 0 10 0 .24 8 75 1 3 .6 8 0 -8 .2 35 7 9 1 5 0 .04 8 619 4 .15 1 80 0 .24 0 86 5 .64 2 0 .08 7 8 0 1 5 0 .04 8 03

1 0 4 .28 0 90 0 .23 3 59 0 .87 4 -13 .68 0 8 0 2 3 0 .04 9 651 1 4 .32 8 30 0 .23 1 04 2 .36 9 -0 .0 36 8 1 2 3 0 .04 9 711 2 4 .36 7 50 0 .22 8 97 1 .28 4 -2 .3 41 8 1 2 3 0 .04 9 541 3 4 .62 8 90 0 .21 6 04 3 .63 0 5 .21 2 8 1 2 5 0 .04 9 661 4 4 .88 1 60 0 .20 4 85 -3 .8 58 2 .83 6 8 2 2 5 0 .04 9 791 5 5 .08 1 70 0 .19 6 78 3 .30 8 -15 .86 0 8 3 3 6 0 .04 9 561 6 5 .21 9 30 0 .19 1 60 -2 .1 83 -8 .0 89 8 3 3 9 0 .04 9 571 7 6 .00 2 20 0 .16 6 60 -5 .0 30 -6 .4 42 8 4 4 1 0 .04 9 851 8 6 .19 7 90 0 .16 1 35 -1 .6 72 0 .00 2 8 4 4 1 0 .04 8 561 9 6 .35 0 40 0 .15 7 47 0 .71 8 5 .52 7 8 4 4 3 0 .04 9 872 0 6 .74 1 80 0 .14 8 33 2 .24 0 1 3 .5 1 0 8 4 5 1 0 .04 9 10

par t i ci pat i onf act or effective mass

Figure 4: Dominant vibration modes

DYNAMIC RESPONSE ANALYSIS WITHOUT AXIAL FORCE FLUCTUATION EFFECT

Nonlinear dynamic response analysis is conducted using flexural moment and curvature relationships ofmembers due to axial force acted by dead load. Results are shown in Fig. 5 to Fig. 10. It is revealed in thesegraphs that response curvatures are within allowable values.

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1. 000E-05

1. 000E-04

1. 000E-03

1. 000E-02

1. 000E-01

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15

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16

( top) member ( bottom)

curv

ature

(1/m

yi el d curvatureal l owabl e curvature ( type‡ T)al l owabl e curvature ( type‡ U)response curvature ( type‡ T)response curvature ( type‡ U)

Figure 5: Response curvature and allowable curvature of pier P1

1. 000E- 05

1. 000E- 04

1. 000E- 03

1. 000E- 02

1. 000E- 01

20

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( top) member (bottom)

curv

ature

(1/m



1. 000E- 05

1. 000E- 04

1. 000E- 03

1. 000E- 02

1. 000E- 01

1. 000E+00

30

00

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1. 000E- 04

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ature

(1/m


Figure 8: Response curvature and allowable curvature of vertical member V1

1. 000E- 04

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1. 000E- 01

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( t op) member ( bot t om)

curv

ature

(1/m

yi el d cur vat ur eal l owabl e cur vat ur e ( t ype‡ T)al l owabl e cur vat ur e ( t ype‡ U)r esponse cur vat ur e ( t ype‡ T)r esponse cur vat ur e ( t ype‡ U)

Figure 9: Response curvature and allowable curvature of vertical member V6

f i 5 6 t & l l bl t

1. 000E-04

1. 000E-03

1. 000E-02

1. 000E-01

90

00

19

00

02

90

00

39

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49

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69

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89

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04

0

( P2 si de) member (P3 si de)

curv

ature

(1/m


Figure 10: Response curvature and allowable curvature of arch ring

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DYNAMIC RESPONSE ANALYSIS CONSIDERING EFFECT OF AXIAL FORCE FLUCTUATION

When effect of axial force fluctuation is considered in dynamic response analysis, nonlinear characteristics (i.e.relationship of flexural moment and curvature) of members change as axial force alters. Generally there are threemethods that can be applied to consider axial force fluctuation [2]. These are given below.1. Flexural moment and curvature are modeled to change according to axial force.2. Effect of axial force fluctuation is directly considered by using fiber models.3. Initially, flexural moment and curvature relationship due to axial force acted by dead load are used to

determine members that yield, then, different flexural moment and curvature relationship due tomaximum and minimum axial force are used to verify effect of axial force fluctuation.

Although methods 1 and 2 directly considers the change in flexural moment and curvature relationship, its actualapplication to seismic design are few and validity of analytic results are difficult to evaluate. Therefore, in thisstudy, method 3 is used.Results are shown in Fig. 11 and Fig. 12. Here, response curvatures are found to be within its correspondingallowable values.

CONCLUSIONS

1. According to eigen analysis of the bridge structure considered herein shows that its first vibration modeis dominant.2. Since the structure shows long period, responses are larger in ground motion Type I than Type II.Responses are within allowable values.3. In nonlinear dynamic response analysis considering axial force fluctuation, responses due to minimumaxial force are larger than that of axial force acted by dead load. Responses are found to be within allowablevalues.4. The structure is vulnerable in top and bottom of pier P1, bottom of pier P3, top of vertical member V1and arch’s springing area near pier P3. However, responses are within allowable values.

0. 000E+00

2. 000E-03

4. 000E-03

6. 000E-03

8. 000E-03

1. 000E-02

1. 200E-02

P1(t

op)

P1(b

ott

om

)

P3

(bott

om

)

V1(t

op)

Arc

hsp

ringin

g(P

3si

de)

member

curv

ature

(1/m

response curvature ofmaxi mum axi al forceal l owabl e curvature ofmaxi mum axi al forceresponse curvature of deadl oad axi al forceal l owabl e curvature of deadl oad axi al forceresponse curvature ofmi ni mum axi al forceal l owabl e curvature ofmi ni mum axi al force

Figure 11: Response curvature and allowable curvature (Type I)

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0. 000E+00

5. 000E- 03

1. 000E- 02

1. 500E- 02

2. 000E- 02

2. 500E- 02

3. 000E- 02

3. 500E- 02

4. 000E- 02

P1(t

op)

P1(b

ott

om

)

P3

(bott

om

)

V1(t

op)

Arc

h

spri

ngin

g(P

3

side)

member

curv

ature

(1/m

r es pons e cur vat ur e of maxi mumaxi al f or ceal l owabl e cur vat ur e ofmaxi mum axi al f or c er es pons e cur vat ur e of deadl oad axi al f or ceal l owabl e cur vat ur e of deadl oad axi al f or cer es pons e cur vat ur e of mi ni mumaxi al f or ceal l owabl e cur vat ur e ofmi ni mum axi al f or c e

Figure 12: Response curvature and allowable curvature (Type II)

REFERENCES

1. Japan Road Association (1996), Design Specifications of Highway Bridges, Part V: Seismic Design.2. Japan Road Association (1998), Materials for Seismic Design of Highway Bridges-Seismic Design

Examples of PC Rigid Frame Bridge, RC Arch Bridge, PC Cable-stayed Bridge, Underground ContinuousWall Foundation, Board Foundation, etc.

seismic design of arch bridges during … design of arch bridges during strong earthquake kiyofumi...

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