Exact uncertainty analysis of the bridge dynamic response during random vehicle crossing by statistical methods

Modeling and simulation of the bridge response during vehicle moving on it is very important in optimum bridge design. There are many researches in this field which some of them deals with studying of the structure and bridge uncertainty effects on the bridge-vehicle dynamic response. Some researchers are dealing with vehicle uncertainty that is uncertainty in mass, sprung, damping, velocity, numbers, incoming time and other mechanical parameters of the vehicles. Other researchers are considering the bridge parameters uncertainty like as module of elasticity, bearing immobility and mass of the bridge. Also in most of the researches, analytical and Monte-Carlo simulations have been used for obtaining the dynamic response of the bridge with considering mentioned uncertainty. However there isn’t any accurate and general statistical study on the bridge response with general uncertainty. In other words, the complete set of uncertainty of the vehicles or bridge has not been applied in modeling and also the exact statistical study of the dynamic response of the bridge has not been applied exactly. Therefore in this paper, after modeling of the bridge-vehicle system with coupled finite element beam and discrete vehicle model, most of the uncertainties of the vehicles have been produced with Gaussian probability distribution function and the statistical parameters of the response have been extracted by Monte-Carlo simulation. The selected model applied in this paper is a discrete model of vehicle with four degree of freedom mass sprung system. Also, the Euler-Bernoulli model was used for bridge and the coupled dynamic equation was extracted using finite element modeling of the beam with Hermitian interpolation function for suitably dividing the vehicle forces, applied on the beam elements nodes. In addition the different road surface using ISO standard was applied on the finite element beam model. One of the major contributions of this paper is considering the type of the vehicle crossing on the bridge as an additional uncertain parameter which is not mentioned in the previous literatures. In other words, three classes of the vehicle that is heavy, semi-heavy and light ones was applied in simulation as uncertain parameters. The mechanical characteristics of the vehicle were derived by CARSIM software and by Monte-Carlo simulation using uniform probability distribution function. Then mentioned parameters was produced and entered in New-Mark simulation steps. With studying on the deflection, velocity, acceleration of the bridge and also considering corresponding upper and lower limit band (confidence interval) and variance of the mentioned parameters, extended results of uncertainties effects have been obtained. It should be mentioned, in this paper an exact statistical test method was used for obtaining the statistical properties of the dynamic response. Also, in different road surface and conditions, the simulation was carried out such that four surfaces of the bridge and three type of the vehicle classes was simulated separately and finally one simulation was done in complete set of the uncertainty. Very detailed and complete results were studied and one of the important results of this paper is reporting the most effective uncertainties on the bridge response. As one of the results, it was found that with increasing the velocity of the vehicle, surface roughness and weight of the vehicle, the uncertainty of the response are increasing. As another important result is that the fundamental frequencies of the bridge is less sensitive to the uncertainties of the vehicles' parameters. Derived results of this paper can be applied in optimum designs of the bridges and also in designing the damage identification methods.