A Methodology For Cable Damage Identification Based On Bending Wave

Cables are widely used as important load-transmission parts in civil engineering structures,especially for long-span flexible bridges,due to light own weight and high load capacity.During the long-term loading under harsh environment,corrosion and fatigue are inevitable,leading to the local damages of cables,which may further affect the safety and the reliability of main structure.It is therefore important to identify local damages during the maintenance of in-service cable structures.The dynamic-based structural health monitoring methods,being among the most popular approaches,have strong rubustness for large amount of information.It is well known,however,that lower-order modal parameters,characterizing the global behavior of structure,are usually not sensitive to local damages.In the present work,a new methodology involving bending wave is therefore developed,which offers opportunities to the local damage identification of cables.First,the dispersion relations of the bending waves are derived from the dynamic equation of cable.From the comparison between the dispersion relations of the Euler-Bernoulli cables,Timoshenko cables and elastic coupled wires,the parametric studies show that cable force,bending stiffness,shear stiffness and the contact relation between wires dominate the modal parameters,bending waves,shear waves and guided waves,respectively.The relationship between wave modes and geometric dimensions is then established.From the proposed strategy for the subdivision of frequency band with regard to wavelength,a theoretical basis for the selection of frequency band is provided for cable damage identification.Based on the known dispersion relation,the cable spectral element matrix,derived from the analytical solution of the frequency-domain dynamic response,is applied for the efficient analysis of bending wave propagation in cables.By this approach,the large amount of computational cost in the high-frequency analysis by the finite element method,caused by the requirement of element subdivision and the restriction of time step,are significantly reduced.On the other hand,the conventional finite element method is still applied for static analysis,taking into account the geometric nonlineary,yielding in the initial state of spectral element model.A two-step FEM-SEM(finite element method-spectral element method)approach is then proposed and the program FESEM is developed,which is suitable for the high-frequency analysis of flexible cable structures.From the numerical examples,the sensitivity of bending wave to local damage is preliminarily verified.By fitting the measured responses to the frequency-domain analytical solution,the fittingerror-based objective function is established for the local axial force identification,with regard to the defined sub-segment of cable.In this approach,a few measurements at the different locations of sub-segment are needed.The influence of boundary conditions,which cannot be disregarded in the conventional modal-based cable force identification,is now solved.On the other hand,the static analytical solutions of cable are derived,with regard to the simplified models with negligible chordwise component of own weight and negligible bending stiffness,respectively.Based on the idenfied axial force of the measured sub-segment,the internal force at an arbitrary location of cable can be obtained by means of analytical model updating.Furthermore,by considering each term of the frequency-domain analytical solution seperately,each frequency response of cable can further be decomposed into evanescent wave components and propagating wave components.For damaged cable model,the analytical solution is expressed by piecewise function,taking the damage interfaces as section nodes.The continuity conditions and the equilibrium states are established,yielding in characteristic equations.The reflection coefficients of evanescent waves and propagation waves are derived.Extensively,the modal solution of finite damaged cable,the characteristic solution of semifinite damaged cable,the force response solution of damaged cable subject to a hammer impact,are derived respectively.From the wave decomposition,it is verified that evanescent wave,being one of the components of cable frequency response,is sensitive to local damage.Lastly,a new methodology for the local damage identification of cable is proposed,based on the estimation of wave components in the defined sub-segment.By applying the known model parameters and dispersion relation,a linear transform algorithm to measured signal is proposed,transforming the measured response from the frequency domain to the wavenumber domain,which corresponds to the coefficient of each wave component.The evanescent wave components at both boundaries of sub-segment is then reconstructed,as well as the reflection coefficients of evanescent waves subject to propagating waves,which can indicate damage location.Based on this,taking the theoretical values of the reflection coefficients for different damage cases as a reference,the damage level can then be quantified.From the above theoretical study,a wave-based methodology is developed,having the potential to the local damage identification of cable structures.The proposed methodology may similarly be extended to the application in other types of structures.