^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

Successive earthquakes of Kocaeli and Duzce within three months indicated that even the survived lifeline structures such as bridges under the former event may have damage or collapse potential under the latter event due to their possible stiffness degradation. It is thus important that a rigorous seismic analysis of such structures should account for the effect of prior earthquake damage. For this purpose, nonlinear seismic analysis of a reinforced concrete bridge structure has been carried out under both single and multiple earthquake ground motions. Behavior and response evaluation of the bridge piers subjected to such motions have been discussed in terms of using both flexure-axial and flexure-shear-axial interaction models. Analytical results show that the stiffness degradation under multiple earthquake ground motions is more pronounced than that under single earthquake ground motion. In addition, comparison of the response without and with shear demonstrates that shear deformation is of significance. The response with shear exhibits the increase in displacement demand and decrease in lateral force carrying capacity, leading to a decrease in energy dissipation capacity. It is concluded that seismic analysis of reinforced concrete bridge structure should account for the effect of multiple earthquake ground motions to assess the demand on such structure properly.

Most of the progress in earthquake-resistant design has been achieved in terms of the observation of damage inflicted by earthquakes. The perception of the way in which bridge responds to an earthquake was dramatically changed by the damage observed after the 1971 San Fernando earthquake. Bridge damage induced by that earthquake clearly demonstrated that the seismic design provisions at that time were inadequate [

Whereas most of the damage analyses for such structures have been conducted in terms of the single ground motion of all components, a few studies have been reported in the literature regarding the seismic response analyses of such structures subjected to multiple earthquake ground motions.

The earthquake of Kocaeli on 17 August 1999 caused severe damage to many reinforced concrete structures [

Studies considering repeated sequence effects were conducted in terms of using single-degree-of-freedom systems [

In addition, multiple earthquake effect was also carried out for reinforced concrete and steel structures. Fragiacomo et al. [

However, most of the previous studies considering repeated earthquakes are related to single-degree-of-freedom systems and building structures, and very few studies are conducted for reinforced concrete bridge structures. This situation has motivated the present work; the main objective of which is to investigate the effects of multiple earthquake ground motions on the stiffness degradation of reinforced concrete bridges, particularly piers.

In addition, axial forces on bridge piers may vary during earthquake excitations due to vertical ground motion. Vertical ground motion components combined with horizontal components may significantly affect the axial force variation [

Galal and Ghobarah [

In line with the above, cumulative earthquake damage due to multiple earthquakes was investigated on a reinforced concrete bridge structure. For this purpose, nonlinear inelastic response analysis of a reinforced concrete bridge was carried out under both single and multiple earthquake ground motions. In addition, hysteretic shear-axial interaction representation was employed to simulate flexure-shear-axial interaction. Subsequently, behavior and response with and without shear for reinforced concrete bridge piers are discussed.

ZeusNL [

In static analysis, both force and displacement loading can be applied with independent values or constrained to vary in proportional ratios. In addition, displacement and acceleration time histories can be applied at the supports. Also availed of is static adaptive strategy where the applied load pattern is varying through the different steps of the procedure. This allows the stiffness degradation and the period elongation of the structures to be accurately described. The solution procedure can be full or modified Newton–Raphson method. Automatic load-step reduction is employed to provide an optimum efficiency, and convergence criteria can be defined in terms of either displacement or force.

For dynamic analysis, the Lanczos algorithm is used for eigenvalue analysis to obtain the required natural frequencies and mode shapes. Time-history analysis is performed through numerically integrating the equation of motion using either the unconditionally stable Newmark family of algorithm or the Hilber–Hughes–Taylor algorithm [

A variety of cross section types are available including steel rectangular solid, hollow, and I-sections, reinforced concrete columns (rectangular and circular) and T-beam sections, and both fully and partially encased composite sections over which a number of monitoring areas are divided, in order to account accurately for the inelastic response of structural members. The layout of steel and both confined and unconfined concrete within the cross section can then be modeled, with the computation of stress resultants being performed automatically.

A detailed description of all available elements and material models in ZeusNL [

Inelastic deformations generated during seismic response are not limited to flexural deformation. Saatciouglu and Ozcebe [

In view of the above, the hysteretic shear models for variable axial forces were developed and implemented in ZeusNL [

Hysteretic shear-axial interaction representation. (a) Quatrilinear symmetric envelope curve. (b) Shear stiffness transitions due to axial force variation.

To describe the inelastic hysteretic shear response, a set of hysteresis rules proposed by Ozcebe and Saatcioglu [

Based on the envelope and a set of shear hysteresis rules, hysteretic shear-axial interaction representation was developed. The basic concept in including the effect of varying axial forces is that the stiffness in the current time step is calculated by introducing appropriate shifts corresponding to the current level of axial force between series of envelope curves derived for constant levels of axial force. This is an equivalent stepwise linear approach. These transitions represent either hardening (increase in stiffness) or softening (decrease in stiffness) of the member due to variation of axial force. Cracking, yielding, and ultimate levels are also shifted in accordance with the axial force variation. The envelop curves are derived for several different levels of constant axial force, which are defined by the user and incorporate three levels in compressive axial force, zero axial force, and two levels in tensile axial force. Such an arrangement enables the user with a proper selection of axial force range of interest. The envelope curve corresponding to an axial force between these prespecified levels is established using extrapolation. The graphical representation of the shear stiffness transition due to varying axial forces is illustrated in Figure

Representative validation of hysteretic shear representation [

In order to investigate the effect of multiple earthquake ground motions on the inelastic response of a bridge, a reinforced concrete bridge structure, severely damaged by the Northridge earthquake of 17 January 1994, is selected. The bridge under consideration is a ramp structure (Collector Distributor 36), which continues on a line close to that of the main freeway, La Cienega-Venice Boulevard sector of the I-10 [

The deck of the ramp structure consists of a three-celled box girder and is carried over the multicolumn bent 5, then over three single-column bents 6, 7, and 8, and over the pier wall of bent 9. In the deck, a movement joint forming a structural hinge is placed just under 5 m away from the bent 6. The presence of this movement joint may allow both relative displacement and rotation to occur between the ends of the deck. The columns of all bents consist of 1219 mm diameter reinforced concrete circular sections. Column longitudinal reinforcement is identical for piers 6, 7, and 8, while less longitudinal reinforcement is employed in the columns of bent 5. Cross sections of three-celled box girder and cross sections of columns and pier wall are shown in Figures

Analytical model of collector distributor 36. (a) Three-celled box girder cross section. (b) Cross sections of the columns and pier wall. (c) Structural model (flexure-axial interaction). (d) Structural model with shear (flexure-shear-axial interaction).

Use is made of MCFT [

Two sets of accelerograms (four in total) are selected for comparative dynamic analyses. One set of accelerograms is Kocaeli (K) and Duzce (D) earthquakes, recorded at the same station, Duzce under the two Turkey earthquakes, Kocaeli of August 1999 and Duzce of November 1999, respectively. The other set of accelerograms is San Fernando (F) and Northridge (N) earthquakes, recorded at Pacoima Dam, San Fernando earthquake of 1971 and the City Hall grounds, Northridge earthquake of 1997, respectively. Whereas the former set may represent the effect of damage accumulation on the structure due to their successive nature in the same area, the latter set can be considered as a benchmark since the two stations are not identical although they are geographically proximate. Nonetheless, it is worth noting that the chosen ground motions are not sufficient enough and rather intuitive. However, since the purpose of the present study is to assess the response of the bridge structure experiencing multiple main shocks recorded at almost identical stations with a time interval, the study can be considered as exploratory comparative analyses on the damage accumulation with and without shear. The peak ground accelerations of all components for the four earthquake ground motions are given in Table

Peak ground acceleration of the selected records.

Station | Peak ground acceleration (g) | ||
---|---|---|---|

Longitudinal | Transverse | Vertical | |

Duzce (Kocaeli EQ) | 0.361 | 0.310 | 0.205 |

Duzce (Duzce-Bolu EQ) | 0.514 | 0.377 | 0.345 |

Pacoima Dam (San Fernando EQ) | 1.174 | 1.139 | 0.686 |

City Hall grounds (Northridge EQ) | 0.346 | 0.883 | 0.232 |

Static analysis of each pier was carried out using the MCFT [

Static lateral force-displacement response of piers.

Static response characteristics.

Parameters | Pier 6 | Pier 7 | Pier 8 |
---|---|---|---|

Applied axial force (kN) | 2395 | 2830 | 3180 |

Lateral force capacity (kN) | 2030 | 1645 | 2065 |

Yield displacement (mm) | 32 | 62 | 32 |

Yield curvature (1/m) | 0.00342 | 0.00489 | 0.00344 |

Nonlinear inelastic time-history analyses were carried out for both flexure-axial only model and flexure-shear-axial interaction model of the bridge structure, depicted in Figures

Comparison of ductility demand under D and K + D earthquakes. (a) Under biaxial input ground motions. (b) Under triaxial input ground motions.

Figures

Shown in Figures

Comparison of ductility demand under N and F + N earthquakes. (a) Under biaxial input ground motions. (b) Under triaxial input ground motions.

Maximum transverse displacement under bi- and triaxial input ground motions (in mm).

Input ground motion | Without shear | With shear | |||||
---|---|---|---|---|---|---|---|

Pier 6 | Pier 7 | Pier 8 | Pier 6 | Pier 7 | Pier 8 | ||

Biaxial | D | 26.19 | 43.72 | 34.73 | 33.28 | 53.11 | 42.59 |

K + D | 29.31 | 48.85 | 39.01 | 36.45 | 57.48 | 44.68 | |

N | 30.62 | 50.52 | 40.67 | 35.68 | 58.05 | 46.80 | |

F + N | 46.84 | 78.70 | 63.77 | 54.00 | 85.94 | 68.39 | |

Triaxial | D | 26.48 | 44.13 | 34.95 | 33.54 | 53.49 | 42.76 |

K + D | 29.33 | 48.88 | 38.97 | 36.47 | 57.57 | 44.57 | |

N | 30.60 | 50.51 | 40.63 | 35.62 | 57.97 | 46.74 | |

F + N | 46.77 | 78.54 | 63.68 | 53.75 | 86.00 | 68.36 |

Further investigation was carried out for the effect of multiple earthquakes in terms of the hysteretic response of piers in the transverse direction. Figures

Transverse hysteretic response of the flexure-axial model subjected to biaxial input ground motions of D and K + D earthquakes. (a) Pier 6. (b) Pier 7. (c) Pier 8.

Transverse hysteretic response of the flexure-axial model subjected to biaxial input ground motions of N and F + N earthquakes. (a) Pier 6. (b) Pier 7. (c) Pier 8.

In order to investigate the effect of multiple earthquakes on the response with shear, comparisons of hysteretic response are made for piers 6 and 8 of the flexure-shear-axial interaction model. Figures

Transverse hysteretic response of the flexure-shear-axial interaction model subjected to triaxial input ground motions of D and K + D earthquakes. (a) Pier 6. (b) Pier 8.

Transverse hysteretic response of the flexure-shear-axial interaction model subjected to triaxial input ground motions of N and F + N earthquakes. (a) Pier 6. (b) Pier 8.

As observed, the response under multiple earthquakes experiences a greater displacement and shows more pronounced stiffness degradation. It is important to note from the response with shear under multiple earthquakes that once single cycle of maximum displacement occurs, the reloading stiffness is significantly reduced in comparison with the stiffness of the first cycle. This indicates that multiple earthquake motions combined with shear affect the energy dissipation capacities of piers, leading to the reduction of stiffness of piers and hence the damage mode.

Comparisons of the response with and without shear are made to investigate the effect of shear on the damage mode of the bridge structure subjected to multiple earthquakes. Figures

Transverse displacement response without and with shear subjected to biaxial input ground motions of multiple earthquakes, K + D.

Transverse displacement response without and with shear subjected to biaxial input ground motions of multiple earthquakes, F + N.

Maximum transverse displacement component for the model with shear (in mm).

Input ground motion | Pier 6 | Pier 7 | Pier 8 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Total | Flexure | Shear | Total | Flexure | Shear | Total | Flexure | Shear | ||

Biaxial | D | 33.28 | 20.68 | 12.70 | 53.11 | 40.83 | 12.28 | 42.59 | 26.04 | 16.55 |

K + D | 36.45 | 20.35 | 16.10 | 57.48 | 41.08 | 16.40 | 44.68 | 28.31 | 16.37 | |

N | 35.68 | 24.32 | 12.36 | 58.05 | 47.05 | 11.00 | 46.80 | 30.34 | 16.46 | |

F + N | 54.00 | 29.22 | 24.78 | 85.94 | 62.25 | 23.69 | 68.39 | 40.88 | 27.51 | |

Triaxial | D | 33.54 | 20.92 | 16.62 | 53.49 | 41.24 | 12.25 | 42.76 | 25.64 | 17.12 |

K + D | 36.47 | 20.11 | 16.36 | 57.57 | 39.07 | 18.50 | 44.57 | 28.44 | 16.13 | |

N | 35.62 | 24.35 | 12.27 | 57.97 | 46.70 | 11.27 | 46.74 | 30.25 | 16.49 | |

F + N | 53.75 | 29.32 | 24.43 | 86.00 | 59.46 | 26.54 | 68.36 | 40.08 | 28.28 |

Figures

Transverse lateral force response without and with shear subjected to biaxial input ground motions of multiple earthquakes, K + D.

Transverse lateral force response without and with shear subjected to biaxial input ground motions of multiple earthquakes, F + N.

Figure

Transverse hysteretic response without and with shear subjected to biaxial input ground motions of multiple earthquakes, F + N. (a) Pier 6. (b) Pier 8.

Maximum transverse lateral force without and with shear under multiple input ground motions (kN).

Pier | Analytical model | Biaxial input motions | Triaxial input motions | ||
---|---|---|---|---|---|

K + D | F + N | K + D | F + N | ||

Pier 6 | Without shear | 1992 | 2249 | 2008 | 2268 |

With shear | 1397 | 1884 | 1461 | 1900 | |

Pier 7 | Without shear | 1254 | 1498 | 1256 | 1490 |

With shear | 1160 | 1594 | 1317 | 1490 | |

Pier 8 | Without shear | 2281 | 2442 | 2289 | 2450 |

With shear | 1934 | 2145 | 1973 | 2163 |

Analytical study has been undertaken for the nonlinear inelastic seismic response of a reinforced concrete bridge under single and multiple earthquake ground motions. The analytical results show that the maximum displacement ductility demand imposed on the bridge piers depends primarily on the applied ground motion characteristics. Although the ductility demand under multiple earthquake motions is greater than that under single motion, the difference between the two cases is not significantly considerable. However, the stiffness degradation under multiple earthquake ground motions is more pronounced than that under single earthquake motion. This is supported by the hysteretic response of piers. The response shows that once the first cycle of maximum displacement is attained, loading stiffness of the second cycle under multiple earthquake ground motions is significantly reduced, in conjunction with a number of inelastic cycles, leading to experience more damage.

The effect of multiple earthquake ground motions on the response with shear is also investigated. Comparisons of the response without and with shear demonstrate that shear deformation is important. The response with shear exhibits the increase in displacement demand, decrease in lateral force carrying capacity, and energy absorption and dissipation capacity of piers. Moreover, shear deformation reaches a significant level (up to 45% to total deformation in this study).

In short, the multiple earthquake ground motions can considerably affect both flexural and shear stiffness degradation and deformation capacity in conjunction with shear. In addition, the response of bridge piers can be affected by the applied earthquake input motion sequence. Thus, the seismic analysis and design of bridges piers should account for the effect of multiple earthquake ground motions in order to assess the demand on such members properly.

This study is a numerical study, and the numerical study results may be sent upon requests.

Parts of figures and analytical results were employed from the paper by the first author, Do Hyung Lee. The paper is “Damage potential of RC bridge piers due to multiple earthquakes,” ^{th}

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT: Ministry of Science and ICT) (no. NRF-2018R1A2B6005716). The authors are grateful for the contribution by the first author.