Vertical vibration of suspension bridges due to traffic and vertical ground acceleration

Bridges are among the important civil infrastructures and are normally designed to have long life spans. Suspension bridges has a larger span in comparison to any other form of bridges. As it get larger span it become more flexible structure. As a structure get flexible its behaviour under dynamic load become more complicated. Suspension bridges are infrastructures in which the load-bearing portion is hung under the suspension cables upon the vertical hangers. Due to their aesthetic, architectural beauty and capability of utilization in long spans suspension bridges, have gained much popularity in recent decades. Throughout the history of suspension bridges, their tendency to vibrate under different dynamic loadings such as wind, earthquake, and traffic loads has been a matter of concern for engineers and researchers. Along with the rapid development of modern transportation networks, suspension bridges are often adopted to wide span rivers or deep valleys. Increasing the construction of suspension bridges and the challenge of modelling their behaviour has attracted the interest of researchers. Many difficulties related to design arises due to its long central span. There are many suspension bridges around the world and dynamic behaviour has been found to be the primary concern for those bridges. Several investigations have been taken place in recent years to determine the behaviour of suspension bridges and dynamical characteristics of it when they are vibrating. However, the complexity of the structure of a suspension bridge makes difficulties on determining dynamical behaviour and dynamical features of suspension bridge. Various studies on dynamic response of moving loads have been conducted on ordinary bridges. However because of complex structures of suspension bridge which consist of various components with different properties, they cannot be directly applied to cable supported bridges. Consequently, more research is required on cable supported bridges to consider the complex structural response and realistically predict their response under moving loads. Various methods of analysis have been applied to the study of the behavior of suspension bridges. One of analytical methods for determining behavior of a suspension bridge is deflection theory. A historical review of the approximate methods that lead to the deflection theory can be found elsewhere. The well-established deflection theory tries to solve the differential equilibrium equation and allows the use of analytical expressions for the solutions. However, explicit analytical solutions are not always possible, and numerical techniques must be used. With the advent of high-speed computer and through the use of numerical methods, major advances in studying the dynamic characteristics of suspension bridges have been achieved. Many efforts have been given to develop a simplified model that can predict the consistency of the responses with detailed model. In recent years, several commercially available finite element software packages have been used by practicing engineers as well as researchers to evaluate the response of suspension bridges for operational traffic, wind and earthquake loads by considering both property and geometry non-linear behavior. Vibration of the suspension bridges are important topic in design of them and it must taken into account in areas where seismic activity is high or in places where a typhoon and a strong wind blowing occurs. For this purpose, displacement response of the suspension bridge when it was subjected to the moving load and ground motions simultaneously investigated in this study. In previous studies static deflection of the suspension bridge, cable tension and also increment in cable tension are investigated. Some of researchers focused on the vibration of the suspension bridge due to the moving load and oscillator, but in most of the studies, the hangers extensibility have neglected. In this research in addition to computing cable tension due to live loads, deflection, velocity and acceleration response of the suspension bridge computed. In this order, at first equation of motion for the deck and the suspension cable of the suspension bridge is derived. Since extensibility of the hangers is considered in this study, a coupled PDE equation must be solved to determine deflection, velocity and acceleration response of the suspension bridge. After derivation of the equation of motion for the cable and the deck, separating variables performed in order to simplify PDE equation to two separate ODE equations which are dependent on time and ordinate of the suspension bridge. To determine deflection of the suspension bridge, behavior of the suspension bridge divided into two parts, first part was static analysis and second part was dynamic analysis. Numerical method has been used to solve static part since it has integral term in equation. Finite difference method as numerical method implemented for static analysis while in dynamic part exact solutions achieved. To verify the written code in MATLAB, obtained result in static analysis from MATLAB compared with achieved result by Choi's article which was published in 2013. Difference of the results were in acceptable range and it indicates that decreasing intervals in FDM method reduces the difference. To this purpose, a suspension bridge which is subjected to moving load and ground motions simultaneously assumed. Three different load cases considered for analyzing. All of load cases were identical in all aspects and the only difference between them was velocity of the moving load. For this purpose written code prepared in MATLAB to determine deflection, velocity and acceleration response of the suspension bridge with use of improved deflection theory. Outcomes indicate that increasing velocity of the moving load would results in increasing deflection of the suspension bridge. Number of modes that considered in determining deflection, velocity and acceleration of the suspension bridge was 50 modes. Obtained result from improved deflection theory compared with results that achieved by deflection theory. Comparison of these results shows that deflection theory is reliable method for determining deflection of the suspension bridge. Hence, in order to determine elongation, axial forces and stresses in hangers, use of improved deflection theory advised. Natural frequency of a suspension bridge mainly dependent on the span and other structural dimensions related to the stiffness. According to the outcomes, the first two vibration modes of the suspension bridge are symmetric and the first anti symmetric mode of the suspension bridge was appeared on third mode. Three different vertical ground motion records were considered in this study in order to determine behavior of the suspension bridge when it was subjected to the moving load and ground motion simultaneously. To prove the validity of the result, FEM commercial software, ABAQUS were used to simulate behavior of the suspension bridge. and when it is subjected to the moving load and after that obtained result from MATLAB was compared with the result that obtained from ABAQUS software. Results shows that achieved outcomes from both of the softwares are in good agreement which proves the reliability of the written code in MATLAB.