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This study consists of the development and presentation of example of seismic isolation system analysis and design for a continuous, 3-span, cast-in-place concrete box girder bridge. It is expected that example is developed for all Lead-Rubber Bearing (LRB) seismic isolation system on piers and abutments which placed in between super-structure and sub-structure. Design forces, displacements, and drifts are given distinctive consideration in accordance with Caltrans Seismic Design Criteria (2004). Most of all, total displacement on design for all LRBs case is reduced comparing with combined lead-rubber and elastomeric bearing system . Therefore, this represents substantial reduction in cost because of reduction of expansion joint. This presents a summary of analysis and design of seismic isolation system by energy mitigation with LRB on bridges.

Seismic isolation systems are the separating of structures (such as bridge, building, railway, road, airport, harbor, dam, and tunnel etc.) from ground motions generated by earthquakes which could induce damage to the structures. Among various seismic isolation systems, lead-rubber bearing (LRB) which has innovative mechanism can lead to increased effective stiffness and is accommodating force in reinforced concrete (RC) structures. LRB is a novel apparatus based on the combination of laminated layers rubber bearing using lead plugs (Constantinou, et al. (2006) [

Figures 1-3 show, respectively, the plan and elevation, the abutment sections and a section at an intermediate bent. In

The bridge is isolated with two multi-directional seismic isolators at each abutment and pier location for a total of eight multi-directional seismic isolators with lead rubber bearings. The isolators are directly located above the circular columns. The plan views of the isolated bridge are shown in

The bridge is isolated with two isolators at each abutment and pier location for a total of eight isolators. The isolators are directly located above the circular columns. The use of two isolators versus a larger number is intentional for the following reasons:

・ With elastomeric bearings it is possible to achieve a larger period of isolation (more mass per bearing).

・ The distribution of load on each isolator is accurately calculated. The use of more than two isolators per location would have resulted in uncertainty in the calculation of the axial load in vertically stiff bearings.

・ Reduction in construction cost.

For better distribution of load to the bearings, diaphragms are included in the box girder at the abutment and pier locations above the isolators. An additional 596 kN weight at each diaphragm location is introduced by the addition of these diagrams. The bridge is considered to have three traffic lanes. Loadings were determined based on AASHTO LRFD Specifications (2001) [

The cross sectional properties of the bridge and weights are presented in

The weight of the seismically isolated bridge superstructure is 22,650 kN. The difference is due to the presence now of diaphragms at the abutment and pier locations in order to transfer loads to the bearings. Kim (2007) [

Element Property | Box Girder | Bent Cap Beam | Column | Rigid Girder | Rigid Column | Rigid Footing |
---|---|---|---|---|---|---|

Area ^{2}) | 6.76 | 2.23 | 1.17 | 18.58 | 18.58 | 18.58 |

Shear Area ^{2}) | 2.25 | 2.23 | 1.17 | 18.58 | 18.58 | 18.58 |

Shear Area ^{2}) | 5.30 | 2.23 | 1.17 | 18.58 | 18.58 | 18.58 |

Moment of Inertia ^{4}) | 82.82 | 0.27 | 0.08^{**} | 854.07 | 854.07 | 854.07 |

Moment of Inertia ^{4}) | 3.42 | 0.61 | 0.08^{**} | 854.07 | 854.07 | 854.07 |

Torsional Constant ^{4}) | 15.12 | 0.64 | 0.21 | 854.07 | 854.07 | 854.07 |

Weight (kN/m) | 207.9^{***} | 76.8 | 27.6 | 0 | 0 | 858.4^{****} |

Constant | K_{X}_{'} (kN/m) | K_{Y}_{'} (kN/m) | K_{Z}_{'} (kN/m) | K_{rX}_{'} (kN-m/rad) | K_{rY}_{'} (kN-m/rad) | K_{rZ}_{'} (kN-m/rad) |
---|---|---|---|---|---|---|

Description | Vertical stiffness | Transverse stiffness | Longitudinal stiffness | Torsional stiffness | Rocking stiffness about Y' | Rocking stiffness about Z' |

Value | 1,391,158 | 1,517,895 | 7,517,895 | 1.57 × 10^{7} | 9.7 × 10^{6} | 9.7 × 10^{6} |

Loading | Abutment Bearings (per bearing) | Pier Bearings (per bearing) | ||
---|---|---|---|---|

Reaction (kN) | Reaction (rad) | Reaction (kN) | Reaction (rad) | |

Dead Load | V + 1497 | 0.00149 | V + 4166 | 0.00006 |

Live Load (Truck, Tandem or Lane) | V + 610 V − 69 | 0.00057 | V + 1101 V − 84 | 0.00040 |

HL93 (Live + IM + BR) | V + 835 V − 119 | 0.00090 | V + 1550 V − 139 | 0.00064 |

Braking (BR) | V + 14 V − 14 | 0.00006 | V + 18 V − 18 | 0.00004 |

Wind on Load (WL) | V + 11 V − 11 T 10 | Negligible | V + 31 V − 31 T 29 | Negligible |

Wind on Structure (WS) | V + 12 V − 12 T 28 | Negligible | V + 40 V − 40 T 90 | Negligible |

Vertical Wind on Structure (WV) | V − 142 | Negligible | V − 458 | Negligible |

Seismic loading is defined per Caltrans Seismic Design Criteria (2004) [

This case study is developed as an alternative to the combined lead-rubber and elastomeric bearing isolation system (Constatinou, et al., 2007 [

Criteria for applicability of single mode analysis are presented in

where (a) for abutment bearing,

Method of Analysis | Application Criteria |
---|---|

Single Mode | 1) Soil profile type A, B, C or D. 2) Bridge without significant curvature, defined as having a subtended angle in plan not more than 30˚. 3) Effective period |

Multimode | 1) Soil profile type A, B, C or D. 2) Bridge of any configuration. 3) Effective period |

Response History | 1) Applicable in all cases. 2) Required when distance to active fault is less than 10 km. 3) Required when soil profile type is E or F. 4) Required when |

Parameter | Upper Bound Analysis | Lower Bound Analysis |
---|---|---|

Maximum Displacement D_{M} (mm)^{1} | 269 | 396 |

Total Maximum Displacement D_{TM} (mm)^{2} | N.A. | 457.2 |

Base Shear/Weight | 0.350 | 0.240 |

Abutment Bearing Seismic Axial Force (kN)^{3} | 813.14 | 784.22 |

Pier Bearing Seismic Axial Force (kN)^{3} | 2820.17 | 2786.81 |

Abutment Bearing Seismic Axial Force (kN)^{4} | 524.00 | 307.37 |

Pier Bearing Seismic Axial Force (kN)^{4} | 1124.96 | 1028.43 |

Effective Stiffness of Each Abutment Bearing K_{eff} (kN/m) | 3730 | 1620 |

Effective Stiffness of Each Pier Bearing K_{eff} (kN/m) | 3730 | 1810 |

Effective Damping | 0.352 | 0.242 |

Damping Parameter B | 1.800 | 1.660 |

Effective Period T_{M} (sec) (Substructure Flexibility Neglected) | 1.76 | 2.58 |

Effective Period T_{M} (sec) (Substructure Flexibility Considered) | 1.87 | 2.67 |

^{1}Based on one-directional excitation in longitudinal bridge direction; ^{2}Based on three-directional excitation using 100% - 30% - 30% rule, and multiplying by Factor 1.1; ^{3}Value is for 100% vertical + 30% transverse + 30% longitudinal combination (maximum axial load); ^{4}Value is for 100% transverse + 30% vertical + 30% longitudinal combination (worst case for lead-rubber bearing safety check-combined with maximum bearing displacement).

In Equation (2),

where

Torsional constant is set

Response spectrum analysis was performed using the response spectrum of _{M}, where T_{M} is the effective period and B is the parameter that relates the 5%-damped spectrum to the spectrum at the effective damping. Quantities T_{M}, B and the effective damping are presented in _{M} are 1.3 sec for upper bound analysis and 2.0 sec for lower bound analysis. Values of parameters used in response spectrum analysis of lead-rubber bearing isolation system are presented in

Eigenvalue and response spectrum analysis were performed in program SAP2000 [

Analysis was performed by separately applying the earthquake excitation in the longitudinal, transverse and vertical bridge directions. The vertical response spectrum was taken as a 70% portion of the horizontal 5%- damped spectrum without any modification for increased damping.

The results of these analyses are presented in _{TM} was calculated as the vectorial sum of bearing displacements due to longitudinal and transverse earthquake components combined using the 100% - 30% rule and then multiplying by Factor 1.1. The results demonstrate very good agreement in the calculated bearing displacement demands and isolation shear forces between the two methods of analysis.

Axial bearing forces are underestimated by the single mode analysis method due, primarily, to neglect of the skew in the calculations. Consideration of the skew angle is not difficult but the underestimation in the calculation of loads to have any significance in the safety of the bearings. Calculations show that the bearings have capacity to sustain the calculated loads in the maximum earthquake.

Bearing Location | Parameter | Upper Bound Analysis | Lower Bound Analysis |
---|---|---|---|

Abutment | Effective Horizontal Stiffness K_{eff} (kN/m) | 3730 | 1620 |

Vertical Stiffness K_{v} (kN/m) | 1,760,000 | 1,760,000 | |

Height h (mm) | 398.8 | 398.8 | |

Modulus E (MPa) | 99,963 | 99,963 | |

Area A (mm^{2}) | 6987 | 6987 | |

Moment of Inertia I (cm^{4}) | 19.5978 | 8.4982 | |

Pier | Effective Horizontal Stiffness K_{eff} (kN/m) | 3730 | 1810 |

Vertical Stiffness K_{v} (kN/m) | 1,760,000 | 1,760,000 | |

Height h (mm) | 398.8 | 398.8 | |

Modulus E (MPa) | 99,963 | 99,963 | |

Area A (mm^{2}) | 6987 | 6987 | |

Moment of Inertia I (cm^{4}) | 19.5978 | 8.4982 |

Period T (sec) | Spectral Acceleration for 5%-Damping (g)^{*} | Spectral Acceleration for Upper Bound Analysis (g) | Spectral Acceleration for Lower Bound Analysis (g) |
---|---|---|---|

0 | 0.70 | 0.70 | 0.70 |

0.05 | 0.70 | 0.70 | 0.70 |

0.10 | 1.29 | 1.33 | 1.33 |

0.24 | 1.77 | 1.78 | 1.78 |

0.30 | 1.80 | 1.80 | 1.80 |

0.50 | 1.72 | 1.75 | 1.75 |

0.75 | 1.44 | 1.44 | 1.44 |

1.00 | 1.19 | 1.19 | 1.19 |

1.25 | 0.95 | 0.95 | 0.95 |

1.40 | 0.86 | 0.48 | 0.86 |

1.50 | 0.78 | 0.43 | 0.78 |

1.60 | 0.72 | 0.40 | 0.72 |

1.75 | 0.64 | 0.36 | 0.64 |

2.00 | 0.55 | 0.31 | 0.55 |

2.10 | 0.53 | 0.29 | 0.32 |

2.20 | 0.50 | 0.28 | 0.30 |

2.50 | 0.41 | 0.23 | 0.25 |

2.75 | 0.37 | 0.21 | 0.22 |

3.00 | 0.32 | 0.18 | 0.19 |

3.25 | 0.29 | 0.16 | 0.17 |

3.50 | 0.25 | 0.14 | 0.15 |

3.75 | 0.23 | 0.13 | 0.14 |

4.00 | 0.20 | 0.11 | 0.12 |

Parameter | Upper Bound Analysis | ||
---|---|---|---|

100% Longitudinal EQ | 100% Transverse EQ | 100% Vertical EQ | |

Maximum Bearing Displacement, D_{M} (mm) | 231.1 (P); 289.6 (A) | 43.6 (P); 48.5 (A) | - |

Isolation Shear/Weight | 0.347 | 0.351 | - |

Bearing Axial Force (kN) | 105.9 (P); 133.9 (A) | 362.1 (P); 391.0 (A) | 3226.3 (P); 1217.9 (A) |

Parameter | Lower Bound Analysis | ||

100% Longitudinal EQ | 100% Transverse EQ | 100% Vertical EQ | |

Maximum Bearing Displacement, D_{M} (mm) | 355.6 (P); 406.4 (A) | 373.4 (P); 398.8 (A) | - |

Isolation Shear/Weight | 0.234 | 0.236 | - |

Bearing Axial Force (kN) | 50.0 (P); 61.8 (A) | 230.0 (P); 251.8 (A) | 3226.3 (P); 1217.9 (A) |

4. Conclusions

The examples presented in this paper demonstrate that single-mode and multi-mode analysis methods, when properly implemented, provide results in close agreement. On the basis of the results obtained in this study, the single mode method of analysis is sufficiently accurate and conservative to be used in analysis and design.

Examples of specifications which are consistent with the assumptions made in the analysis have been presented in this study.

System | Lower Bound Analysis | Upper Bound Analysis | ||||||
---|---|---|---|---|---|---|---|---|

D_{TM} (mm) | D_{M} (mm) | |||||||

Lead-Rubber and Elastomeric Bearing^{*} | 533 | 0.250 | 0.098 | 0.152 | 318 | 0.308 | 0.095 | 0.213 |

Lead-Rubber Bearing | 457 | 0.240 | 0.113 | 0.127 | 269 | 0.350 | 0.175 | 0.175 |

1) Total displacement

2) In lower bound analysis, the benefit of all LRB versus Elastomeric bearing/LRB is almost 2% reduction in shear force at pier. When

3) Furthermore, while previous design requires tests for two types of bearing, another benefit is that construc- tion and testing is needed to only one type of bearing.

The authors would like to gratefully acknowledge Professor Michael C. Constantinou for his advice to this study. This research was supported by research fund, Kumoh National Institute of Technology.