SEISMIC DESIGN OF ARC BRIDGES DURING STRONG EARTQUAKE Kiyofumi NAKAGAWA 1, Tatsuo IRIE 2, Allan D SUMAYA 3 And Kazuya ODA 4 SUMMARY In structural design of arch bridges, it is essential to determine plastic regions generated during strong earthquakes and to consider the effect of axial force fluctuation in dynamic response analysis. In this study, nonlinear time history analysis is performed with respect to bridge s axis and transversal axis where the bridge is modeled as two-dimensional frame model. Three long periodic and three short periodic acceleration waves of approximately 3.30 m/sec2 and 6.30 m/sec2 maximum, respectively, are considered as seismic loads. The natural period in the bridge axis and transversal axis directions are found to be 1.47 sec and 1.17 sec, respectively, which implies that long 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 arch ring when seismic force acts in bridge axis. In the transversal axis, plastic regions occur at the base of 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 produced even in short periodic earthquakes but safety of the bridge is still ensured since the ductility factor obtained is less than three. Comparing the results of the case where axial force fluctuation is considered with the case where it is neglected, curvature at the base of arch ring increases by 20 percent and sectional force at the base of pier by 10 percent. INTRODUCTION The bridge considered in this study is a concrete deck anger arch bridge with prestressed stiffening girders. It is 300 meters long with 143-meter arch span, which will be constructed over the Takachiho ravine, one of Japan s quasi-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; and 3. Economical value of the bridge. Different types of bridges that satisfy the above conditions were compared and evaluated. Overall evaluation result shows that reinforced concrete anger arch bridge with prestressed stiffening girders is the most appropriate structure. Seismic design of this bridge for strong earthquakes is performed by nonlinear dynamic analysis. The method of analysis and computed results along bridge axis is presented in this paper. 1 2 3 4 CTI Engineering Co., td., Fukuoka, Japan,Email: k-nakagw @fukuoka.ctie.co.jp CTI Engineering Co., td., Fukuoka, Japan,Email: irie @fukuoka.ctie.co.jp CTI Engineering Co., td., Fukuoka, Japan, Email: allan@fukuoka.ctie.co.jp Ministry of Construction, Japan
SUMMARY OF BRIDGE a) Type of bridge: Reinforced concrete anger arch bridge with prestressed stiffening girders b) Bridge length: =300 meters c) Span: l = 2@40 m + 150 m + 2@35 m d) Arch span: 143 meters e) Road width: W=15.5 meters f) Cross section: refer to Fig. 1 g) General view of bridge: refer to Fig. 2 P U R O O S O O S O O O V T O O S O O O S O O R T O O T O O R V T O R V T O T O O R T O O Q T O O Q Q T O Q T O P O O 2. 0 Q O O 1.5 Q O O P O O 1.5 2. 0 Q O O R S T O T O O W O O O T O O R S T O Q O O Figure1: Cross section of box girder R O O @ O O O S O @ O O O S O @ O O O P T O @ O O O R T O O O R T O O O U O @ O O O R O @ O O O U O O O O P R O O O W O O O ` 1 Q U O O O P O O O O o 1 T P O O O R X R O O P S O O O Q T O O V T O V T O Q @ T O O S @ O O O S @ O O O P T O O O V O O O P O O O b b o @ƒó T T O O ` ` 1 k V. O O O @ m Q W @ T O O Œ P O O O ` ` 2 P T O O O o 3 S O O O O P O O O O Q O O O V T O O o 4 P T T O O R O O O Œ S O O O ` 2 X O O O o 2 P S R O O O ƒ ˆ Œ @ ƒ ƒ ˆ Figure2: Side view of bridge METOD OF ANAYSIS Arch bridges, according to Design Specifications for ighway Bridges-Part V Seismic Design (1996) [1], are considered as structures that exhibit complex behaviour during strong earthquakes. Moreover, since it also statically indeterminate structures of high degree, nonlinear structural members are expected to generate in more than one area. This suggests that, in this case, energy constant rule is not valid and ductility design method is inapplicable. 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 fluctuate due to horizontal action of earthquake. Effect of axial force fluctuation suggests changing of flexural moment and curvature relationship in the analysis. In this study, the effect of axial force fluctuation is investigated by comparing response curvature when axial force is maximum and minimum with allowable curvature. ere, relationship of flexural moment and curvature is varied in structural members found to yield when flexural strength due to dead load is considered. There are two types of ground motions used in the analysis as indicated in Design Specifications of ighway Bridges-Part V Seismic Design (1996) [1], that is, Type I and Type II. Type I corresponds to plate boundary type large-scale earthquakes, which characterized by long periodic waves. On the other hand, Type II corresponds to inland direct strike type earthquake like the 1995 yogo-ken Nanbu Earthquake, which characterized by short periodic waves. Also, both are ground motions with high intensity, though less probable to occur during the service period of the bridge. Acceleration spectra for strong grounds (i.e. classified as ground Type I in Design 2
Specification of ighway Bridges), acceleration spectra ranges from 2.0 to 7.0 m/sec 2 in ground motion Type I and 0.75 to 200 m/sec 2 in Type II. CONDITIONS FOR ANAYSIS 5. 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 ring inear beam elements: pier (P4), vertical member (V2 to V5), stiffening girder along bridge axis 1 2 3 4 5 6 789 10 11 12 13 14 15 16 17 18 19 21 22 23 25 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 20 24 28 4001 5001 6001 301 401 402 101 201 1001 2001 3001 6004 303 104 203 1004 3003 9019 9021 9022 9024 V4 4003 5002 6005 304 404 V3 P4 A2 204 1005 2002 4005 6006 405 3005 5003 305 6007 406 106 1006 9016 90179018 9026 9027 V5 A1 306 408 107 205 2003 V2 1007 5004 6008 9015 9029 307 108 206 90309031 1008 5005 6009 9014 V6 109 P1 2004 308 207 V1 9013 9032 6010 110 1009 2005 6011 P3 9012 111 6012 309 9033 112 208 1010 6013 P2 9011 9034 6014 310 114 1011 9035 6017 117 209 9010 9037 311 1012 9038 1013 312 9009 210 1014 9039 313 1017 9008 314 9006 9040 315 211 317 9005 9042 212 213 9004 214 9003 215 217 9001 R R R R R R R S R : rigid : hinge : linear member : non linear member S : slide Figure 3: Frame model used in the analysis e) Strength of structural members: refer to Table 1 f) Relationship of flexural moment and curvature for nonlinear structural members: Tri-linear degrading stiffness model (Takeda model) considering strength of reinforced concrete during cracking, yielding and ultimate stage. g) Damping constants: inear structural members: h = 0.05 Nonlinear structural members: h = 0.0.2 Strain energy types of modal damping constants are calculated using these values. Also, Rayleigh damping 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 zero acceleration 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. 3
Table 1: Strength of structural members Superstr ucture Substruc ture Structural Concrete Steel bars Arrangement of reinforcement bars notes Stiffening 40 /mm 2 SD295 Arch ring 40 /mm 2 SD345 Main bars: D32ctc125(2.0steps), Vertical 30 /mm 2 SD345 Main bars: D32ctc125(1.0step), V1, V6 member 24 /mm 2 SD295 V2 to V5 Pier P1 30 /mm 2 SD345 Main bars: D51ctc150(2.0steps), Pier P2 21 /mm 2 SD295 Main bars: D38ctc125(1.0step), Pier P3 21 /mm 2 SD295 Main bars: D51ctc150(1.0step), Pier P4 21 /mm 2 SD295 6. EIGEN ANAYSIS 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 found to be dominant. Table 2: Eigen Analytic Results degree of mode natural frequency natural period i (z) (sec) participation factor al ong br i dge axi s vert i cal di r ect i on al ong br i dge axi s modal damping constant vert i cal h di r ect i on i effective mass 1 0.67875 1.47330 36.960 1.684 61 0 0.04562 2 1.42730 0.70062 1.031-1.091 61 0 0.04980 3 2.21750 0.45097 5.129-0.935 62 0 0.04958 4 2.65970 0.37598-2.053 1.191 62 0 0.04911 5 3.18250 0.31422 3.455-11.270 63 6 0.04980 6 3.80090 0.26309-11.470-1.160 68 6 0.04962 7 3.89090 0.25701 7.296 11.030 71 12 0.04928 8 4.02010 0.24875 13.680-8.235 79 15 0.04861 9 4.15180 0.24086 5.642 0.087 80 15 0.04803 10 4.28090 0.23359 0.874-13.680 80 23 0.04965 11 4.32830 0.23104 2.369-0.036 81 23 0.04971 12 4.36750 0.22897 1.284-2.341 81 23 0.04954 13 4.62890 0.21604 3.630 5.212 81 25 0.04966 14 4.88160 0.20485-3.858 2.836 82 25 0.04979 15 5.08170 0.19678 3.308-15.860 83 36 0.04956 16 5.21930 0.19160-2.183-8.089 83 39 0.04957 17 6.00220 0.16660-5.030-6.442 84 41 0.04985 18 6.19790 0.16135-1.672 0.002 84 41 0.04856 19 6.35040 0.15747 0.718 5.527 84 43 0.04987 20 6.74180 0.14833 2.240 13.510 84 51 0.04910 Figure 4: Dominant vibration modes DYNAMIC RESPONSE ANAYSIS WITOUT AXIA FORCE FUCTUATION EFFECT Nonlinear dynamic response analysis is conducted using flexural moment and curvature relationships of members due to axial force acted by dead load. Results are shown in Fig. 5 to Fig. 10. It is revealed in these graphs that response curvatures are within allowable values. 4
1. 000E- 01 al l owabl e curvat ure ( t ype T) al l owabl e curvat ure ( t ype U) 1. 000E- 05 100001 1. 000E- 01 100002 100003 100004 100005 100006 100007 100008 100009 100010 100011 100012 100013 100014 100015 Figure 5: Response curvature and allowable curvature of pier P1 100016 1. 000E- 05 200001 1. 000E+00 200002 200003 200004 200005 200006 200007 200008 200009 200010 200011 al l owabl e curvat ure ( t ype T) al l owabl e curvat ure ( t ype U) Figure 6: Response curvature and allowable curvature of pier P2 200012 200013 200014 200015 200016 1. 000E- 01 allowabl e curvat ure ( t ype T) allowabl e curvat ure ( t ype U) 1. 000E- 05 300001 300002 300003 300004 300005 300006 300007 300008 300009 300010 300011 300012 300013 300014 300015 Figure 7: Response curvature and allowable curvature of pier P3 300016 5
1. 000E- 01 10001 10002 10003 10004 10005 10006 10007 10008 10009 10010 10011 allowabl e curvat ure ( t ype T) allowabl e curvat ure ( t ype U) Figure 8: Response curvature and allowable curvature of vertical member V1 1. 000E- 01 10012 10013 10014 10015 10016 60001 60002 60003 60004 60005 60006 60007 60008 60009 60010 60011 yi el d cur vat ur e al l owabl e curvat ure ( t ype T) al l owabl e curvat ure ( t ype U) respons e cur vat ur e ( t ype U) Figure 9: Response curvature and allowable curvature of vertical member V6 60012 60013 60014 60015 60016 1. 000E- 01 allowabl e curvat ure ( t ype T) allowabl e curvat ure ( t ype U) 90001 90002 90003 90004 90005 90006 90007 90008 90009 90010 90011 90012 90013 90014 90015 90016 90017 90018 90019 90020 90021 90022 90023 90024 90025 90026 90027 90028 90029 90030 90031 90032 90033 90034 90035 90036 90037 90038 90039 90040 (P2 side) member (P3 side) Figure 10: fi Response 5 6curvature and allowable t & curvature ll blof arch ring t 6
DYNAMIC RESPONSE ANAYSIS CONSIDERING EFFECT OF AXIA FORCE FUCTUATION 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 three methods 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 to maximum 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 actual application to seismic design are few and validity of analytic results are difficult to evaluate. Therefore, in this study, method 3 is used. Results are shown in Fig. 11 and Fig. 12. ere, response curvatures are found to be within its corresponding allowable values. CONCUSIONS 1. According to eigen analysis of the bridge structure considered herein shows that its first vibration mode is 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 minimum axial force are larger than that of axial force acted by dead load. Responses are found to be within allowable values. 4. The structure is vulnerable in top and bottom of pier P1, bottom of pier P3, top of vertical member V1 and arch s springing area near pier P3. owever, responses are within allowable values. 1. 200E- 02 8. 000E- 03 6. 000E- 03 4. 000E- 03 response curvature of maxi mumaxi al force al l owabl e curvat ure of maxi mumaxi al force response curvature of dead load axial force al l owabl e curvat ure of dead load axial force response curvature of mi ni mumaxi al force al l owabl e curvat ure of mi ni mumaxi al force 2. 000E- 03 0. 000E+00 P1(top) P1(bottom) P3(bottom) V1(top) Arch springing(p3 side) member Figure 11: Response curvature and allowable curvature (Type I) 7
4. 000E- 02 3. 500E- 02 3. 000E- 02 2. 500E- 02 2. 000E- 02 1. 500E- 02 response curvature of maxi mum axi al f orce al l owabl e cur vat ur e of maxi mum axi al f orce response curvature of dead load axial force al l owabl e cur vat ur e of dead load axial force response curvature of mi ni mum axi al f orce al l owabl e cur vat ur e of mi nimum axi al f orce 5. 000E- 03 0. 000E+00 P1(top) P1(bottom) P3(bottom) V1(top) Arch springing(p3 side) member Figure 12: Response curvature and allowable curvature (Type II) REFERENCES 1. Japan Road Association (1996), Design Specifications of ighway Bridges, Part V: Seismic Design. 2. Japan Road Association (1998), Materials for Seismic Design of ighway Bridges-Seismic Design Examples of PC Rigid Frame Bridge, RC Arch Bridge, PC Cable-stayed Bridge, Underground Continuous Wall Foundation, Board Foundation, etc. 8