LITERATURE REVIEW
2.4 Dynamic Analysis
All real physical structures behave dynamically when subjected to loads or displacements. The additional inertia forces, from Newton's second law, are equal to the mass times the acceleration. Ifthe loads or displacements are applied very slowly, the inertia forces can be neglected and a static load analysis can be justified. Hence, dynamic analysis is a simple extension of static analysis.
According to Mullen and Swann (2001) all real structures having infinite number of displacements have the difficulty to create a computer model with a finite number of mass-less members and a finite number of node displacement that will simulate the behaviour of the real structure. The mass of a structural system, which can be accurately estimated, is lumped at the nodes. However, the dynamic loading, energy dissipation properties, and boundary (foundation) conditions for many structures are difficult to estimate. This is always true for the cases of seismic input or wind loads.
To reduce the errors that may be caused by the approximations summarized in the previous paragraph, it is necessary to conduct many different dynamic analyses using different computer models, loading and boundary conditions.
2.4.1 Mode superposition method
The mode superposition method is the most common and effective approach for seismic analysis of linear structural systems. All types of loading can be accurately approximated by piece-wise linear or cubic functions within a small time increment.
After a set of orthogonal vectors have been evaluated, this method reduces the large set of global equilibrium equations to a relatively small number of uncoupled second order differential equations. The numerical solution of those equations involves greatly reduced computational time. (SAP, 2000)
Modal superposition provides a highly efficient and accurate procedure for performing time-history analysis. Closed-form integration of the modal equations is used to compute the response, assuming linear variation of the time functions,
f
i(t), between the input data time points. Therefore, numerical instability problems are never encountered. (SAP, 2000)2.4.2 Previous studies ou dynamic aualysis of bridges
Behaviour of bridges under the influence of seismic load has been a major point of interest for engineers over a long period of time. The 1971 San Fernando earthquake was a major turning point in development of seismic design criteria for bridges.
Although significant advances have been ac~ieved since that t~e in the design and construction of an earthquake resistant bridge, numerous gaps still remain in the understanding of the seismic behaviour of bridges.
Kunnath and Gross (1999) presented the inelastic damage evaluation of a typical double-deck bent of the Cypress Viaduct which collapsed during the 1989 Lorna Prieta earthquake. They developed a model of the bent consisting of spread plasticity- based beam-column elements to represent the piers and the deck, and shear panel elements to represent the pedestal region. Also an element fibre model was conducted to determine beam and colunrn moment -curvature relationships accurately. In addition, a smeared-crack approach finite element analysis was employed to determine the lateral load-deformation relationship of the pedestal regions. The model of the Cypress Viaduct was subjected to the Oakland Outer Harbor Wharf ground acceleration record in the plane of the bent. The analytical model was calibrated using static lateral load tests, ambient and forced vibration tests, and observed performance.
The results of time-history analyses, which include a prediction of member damage, indicate that collapse was initiated by a shear failure of the pedestal regions.
Both the serviceability and safety of bridge structures dynamic response during earthquake is a very important factor. The controlling parameters that govern dynamic response of a bridge depend on different structural attributes of a particular bridge.
Chaudhury at el. (2000 and 2002) identified the system parameters of base-isolated bridges with the help of records made on a base-isolated bridge during a strong
earthquake. Tan and Huang (2000) also developed an identification algorithm to investigate dynamic properties of base-isolated highway bridges equipped with lead- rubber bearings. A number of schemes for identification of dynamic parameters of bridges have been developed in recent years. Most of the schemes are, however, for example girder bridges of three to four straight spans.
Kappos et al (2002) investigated the effect of a modelling approach, also including the interaction phenomenon between supporting ground and the pier plus deck system, on the seismic response of reinforced concrete (RIC) bridges with irregular configuration, as well as its ramifications on the design of the piers. The bridge and its foundation system, including the surrounding soil, are modelled by finite elements plus the spring/dashpotJadded mass discrete parameter system. A hierarchy of finite element meshes is developed, starting with shell elements, and ending with linear elements whose performance as far as dynamic loads are concerned is gauged to be completely satisfactory. They gave a series of recommendations on when and how to account for the influence of the ground in the design of the piers.
Lee et al (2005) presented a hysteretic shear-axial interaction model. Their model is capable of representing the shear stiffness transitions due to axial force variation and is implemented in a nonlinear finite element program. A comparative study has been carried out by them for reinforced concrete column tests which shows that the predicted response obtained with the new formulation agrees well with the test results in terms of strength, stiffuess and deformation characteristics. The evaluation of displacement response for piers reveals that displacement is significantly increased due to the effect of shear, thus increasing the imposed ductility demand, leading to overall stiffness degradation and period elongation. This explains the damage pattern ofthe piers more comprehensively than the case without shear.
Zhihao et al (2005) presented a seismic performance upgrading approach for steel arch bridges using buckling-restrained braces as dampers against longitudinal directional earthquake motions. Inelastic behaviour of a representative steel arch bridge with buckling-restrained braces is investigated by three-dimensional modelling and time-history analyses. The results are compared with those from the original structure. They found that replacement of diagonals of some parts by buckling-
restrained braces can greatly improve seismic performances of the steel arch bridge.
As a result, this approach is believed to be an effective way for seismic performance improvement of new bridge designs as well as retrofit of existing ones.
A model of the Jamuna Bridge was developed with the finite element software SAP2000 (Rahman, 2008a). The model, however, did not consider the presence of actual geometry of the piers and curvature of the bridge. Ahsan et al (2005) identified the dynamic parameters of the Jamuna Multipurpose Bridge in ambient transverse vibration. A FE model of Jamuna Multipurpose Bridge was established. All the features of bridge are included in this model such as bridge horizontal curvature, the curvature of the deck etc. A modal analysis was done to observe the bridge behaviour.
Mathematical models are derived for better understanding of dynamic parameters of the bridge. Formulations for single and two degrees of freedom systems were presented to compare dynamic parameters with ambient vibration of the bridge.
Soneji and Jangid (2007) presented the performance of passive hybrid control systems for the earthquake protection of a cable-stayed bridge under real earthquake ground motion. A simplified lumped mass finite-element model of the Quincy Bay-view Bridge at Illinois is used for the investigation. They used a viscous fluid damper (VFD) as a passive supplemental energy dissipation device in association with elastomeric and sliding isolation systems to form a passive hybrid control system. The effects of non-linear viscous damping of the VFD on the seismic response of an isolated cable-stayed bridge are examined by taking different values of velocity exponent of the damper. The seismic response of the bridge with passive hybrid systems is compared with the corresponding response of the bridge with only isolation systems, as well as with the uncontrolled bridge. The results of the investigation show that the addition of supplemental damping in the form of a viscous fluid damper significantly reduces the earthquake response of an isolated cable-stayed bridge. The nonlinear viscous damping is found to be more effective in controlling the peak displacement of the isolated bridge while simultaneously limiting the base shear in towers.
Rahman (2008a) identified the dynamic parameter of Jamuna Multipurpose Bridge.
The amplitudes of bridge response due to the application of earthquakes of differing
magnitudes were evaluated in that study. For that purpose, dynamic parameters were identified by ambient vibration assuming simplified geometry and deflection parameters of the bridge. Finally, a multi degree of freedom system was formulated to determine a Transfer Ratio function of the system. The concept of Transfer Ratio (TR) allows one to use the actual recorded data to determine the system behaviour without any simplified assumption regarding the geometry of the bridge. Using the earthquake data. recorded by the sensors a Transfer Ratio function was derived and used to predict possible response of the Jamuna Bridge. This Transfer Ratio was verified and also applied for the earthquakes of February 14 and August 5, 2006 recorded near the Jamuna Bridge. A detailed finite element model of the bridge considering all features of the bridge geometry such as horizontal and vertical curvature of the bridge, variable deck thickness etc was developed. For this study eight different types of models are developed. A comparative study of bridge response was carried out at various locations under various dynamic loads such as ambient vibration, traffic vibration, combined train and traffic vibration, only train vibration and earthquake. From the responses of bridge under various dynamic loading, it was observed that there was a common peak frequency at around 1.013Hz which almost matches with the predominant frequency in transverse vibration as obtained from the finite element model (1.00174 Hz) and the Transfer Ratio (1.0098Hz).
Rahman (2008b) worked on the results of a Finite Element model of the Jamuna Bridge due to an applied earthquake with the actual data found from the sensors located on the bridge site. The earthquake data were recorded on 17'h June 2004. The actual data of the earthquake as recorded at the Bridge West-End FFS of the Jamuna Multipurpose Bridge is used for obtaining the response of the bridge model. The actual data were recorded in three directions i.e., North- South, East-West and UP- Down. The data of west-end side was given as input to the bridge model. The response of the bridge model is then compared with the actually recorded response of the bridge. The study also attempts to find the applicability of the model to predict the bridge response due to earthquakes and the need to update the model so that it can better reflect the measured data from the physical structure being modelled. Prediction of response of the Jamuna Multipurpose Bridge due to different earthquakes was also an objective. Study of bridge response due to vehicular and train loading with a focus
on their weight and speed has also been carried out. Time history analysis has been performed to determine the effectiveness ofthe model and the prediction of response.