Damage Assessment of Civil Engineering Structures by the Observation of Non-linear Dynamic Behaviour

The detection of damages in civil engineering structures and bridges in particular is mainly done by visual examination. However, defects as for instance partial rupture of a prestressing cable or fatigue cracks in reinforcement can not be visually observed. It is well known that damage changes dynamic structural parameters like eigenfrequencies, eigenmodes and damping. However, the sensitivity to small damages is sometimes low. Therefore, as an alternative the occurrence and evaluation of non-linear dynamic behaviour is considered. The basic idea is that non-linear dynamic effects increase with growing cracks under forced excitation. The implementation of this idea in the regular inspection program of bridges presupposes exact knowledge of the eigenfrequencies of the undamaged structure that are also supposed to be force dependent. This paper presents the results of an experimental approach with three reinforced concrete beams of different damage states investigating the non-linear behaviour due to the excitation force. 2 THE BASICS OF NON-LINEARITY Assuming linear behaviour of a harmonically excited structure, the differential equation (1) describes the vibration adequately, where M is the mass matrix, C the damping matrix, K the stiffness matrix and F the excitation force. ) ( ) ( ) ( ) ( t F t Ky t y C t y M = + + & & & (1) Disturbances in the structure like cracks lead to non-linear stiffnessand damping matrices. In this case the coefficients in the equation of motion are dependent on the vibration amplitude, velocity and thus on the excitation force. Worden et al. (2001) and Dimitriadis et al. (2006) present an overview of the most studied types of nonlinearities. Much research is driven by the aerospace sector. The methodology is applicable to other domains of engineering. A distinction has to be made between nonlinearities depending on displacement and nonlinearities depending on velocity. Figure 1 shows common nonlinearities and their dependence. Figure 1. Common nonlinearities and their dependence on displacement and velocity There is a large amount of different nonlinearities discussed in literature. The abovementioned nonlinearities are representative for many different types occurring in vibratory systems. They allow a theoretical and approximate description of non-linear behaviour. Nonlinearities may occur, for which a classification according to the types in figure 1 is not possible. In this case a description can be based on a combination of nonlinearities or on a nonlinear material behaviour. 3 HOW TO DETECT NON-LINEARITIES IN MEASUREMENTS 3.1 Is the systems vibration non-linear? For the investigation of nonlinear dynamic behaviour it is initially essential to discover, whether the structure behaves nonlinear at all. For this purpose four elementary operations are presented. For linear systems the principle of superposition is valid. This principal applies both to static systems and to dynamic systems. It reads as follows: The motions of a body proceeding at the same time do not affect each other mutually. The resulting quantities (displacement, velocity, acceleration) arise from a geometrical addition of the components. ) ( ) ( ) ( ) ( ) ( ) ( ); ( ) ( 2 1 2 1 2 2 1 1 t x t x t y t y t x t y t x t y + → + ⇒ → → (2) Superposition is independent of the type of excitation. If this principle does not apply, nonlinear dynamical behaviour exists. As well the distortion (harmonic distortion) of the response vibration, excited by a sinusoidal excitation force, is an explicit indicator for the existence of nonlinearity. In the case of linear vibration the ratio of the amplitudes of the input (X(ω)) and output (Y(ω)) signals is always constant. Hence the frequency response function (FRF) is independent of the excitation force level and can be expressed as: Displacement Velocity Cubic stiffness Bilinear stiffness Hysteretic stiffness Friction Quadratic damping