On damage detection by continuous dynamic monitoring in wind-excited suspension bridges

Abstract

The paper discusses the application of dynamic methods for damage detection in the main cables of suspension bridges, using data continuously recorded under wind excitation through permanent monitoring systems and automated operational modal analysis.

A continuum model for predicting the vertical aeroelastic response of wind-excited damaged suspension bridges is formulated and presented at first. The model shows that, for a real sample bridge, typical variations of mean wind speed produce variations of natural frequencies, due to aeroelastic effects, that are more significant than those produced by a small damage. A possible solution to this issue, proposed in the paper, consists of removing the dependence on the excitation source by calculating frequency shifts considering frequencies, in reference and damaged states, associated to approximately the same mean wind speed. This task and the necessary estimation of frequency shifts through a statistical analysis of identified natural frequencies outline the need for a continuous dynamic monitoring.

The analytical model is finally employed for generating dynamic wind response data that are successively processed by means of an advanced automated modal identification tool. Although based on the simplifications inherently contained in the analytical model, the results show that frequency shifts caused by a relatively small damage can be accurately estimated from response data recorded under wind excitation with a reasonable number of data sets.

Appendices

Appendix A: Structural eigenmodes

The eigenmodes derived in [21] are here recalled for the sake of completeness.

In the case of antisymmetric motions, the eigenfunctions and eigenfrequencies are

$$ \begin{aligned}[c] &\phi_i(\overline{x})=C_i \sin (2i\pi\overline{x}) \\ &\overline{\omega}_i=2\pi i\chi\sqrt{1+ \biggl(2\pi i \frac{\mu}{\chi} \biggr)^2} \end{aligned} $$

(57)

where i=1,2,3,… is the modal order and C _i is a set of amplitude coefficients. In the case of symmetric motions the eigenfunctions are, instead, given by the following equation:

(58)

where $\overline{h}_{i}=\int_{0}^{1}\phi_{i}(\overline{x})\mathrm{d}\overline {x}$, while $\alpha_{i}^{2}$ and $\beta_{i}^{2}$ are defined as

$$ \begin{aligned}[c] & \alpha_i^2= \frac{\chi^2 (\sqrt{1+ (2\overline{\omega}_i\frac{\mu }{\chi^2} )^2}-1 )}{2\mu^2} \\ & \beta_i^2=\frac{\chi^2 (\sqrt{1+ (2\overline{\omega}_i\frac{\mu }{\chi^2} )^2}+1 )}{2\mu^2} \end{aligned} $$

(59)

The unknown $\overline{\omega}_{i}$ are solutions of the following characteristic equation:

(60)

whose calculation requires a numerical procedure.

Appendix B: Generation of buffeting loads

A generation formula for the modal components of the buffeting loads, vector in (40), is here presented using a classical model for the turbulent wind field.

Let us assume that $\overline{u}(\overline{x},\overline{t})$ and $\overline{w}(\overline{x},\overline{t})$ are realizations of zero mean Gaussian random processes and that the cross power spectral density function of u (analogous expressions can be written for w) is given by

$$ S_{u}\bigl(x,x',\omega\bigr)=\overline{S}_{u}( \omega) \operatorname{Coh}_{u}\bigl(x,x',\omega\bigr) $$

(61)

where $\overline{S}_{u}(\omega)$ is the power spectral density of the random process for x=x′ and $\operatorname{Coh}_{u}$ is the so-called coherence function. The following analytical expressions for $\overline{S}_{u}(\omega)$ and $\overline{S}_{w}(\omega )$ can be used [27]:

(62)

(63)

where $\sigma^{2}_{u}$ and $\sigma^{2}_{w}$ are the variances, while L _u and L _w are the integral length scales, of u and w, respectively. The coherence function assumes the classic exponential form

$$ \operatorname{Coh}_{u}\bigl(x,x',\omega\bigr)=\exp \biggl(-\frac{|\omega |C_{ux}|x-x'|}{2\pi U} \biggr) $$

(64)

where C _ux is a positive coefficient whose value depends upon the terrain conditions (a similar expression can be written for $\operatorname{Coh}_{w}$, with exponential coefficient C _wx ).

By subdividing the frequency interval $[0\ \widetilde{\omega}]$ into N _ω steps of amplitude Δω, expressing frequencies in non-dimensional form (11), and posing $\overline{\omega}_{j}=j\varDelta \overline{\omega}$, realizations of the turbulent wind velocities can be obtained by means of the following Monte Carlo formula [27]:

(65)

(66)

where

(67)

(68)

$R_{uj}^{r}$, $I_{uj}^{r}$, $R_{wj}^{r}$ and $I_{wj}^{r}$ being zero-mean normal random numbers with variances equal to 1/2.

In (65), (66) γ _ur and θ _ur are the r-th eigenvalue and eigenfunction of the stochastic process $u(\overline{x},\overline{t})$, while γ _wr and θ _wr are the same quantities for the vertical component $w(\overline {x},\overline{t})$. Using (61), (64), they can be analytically expressed as follows [27] (similar expressions hold for w):

(69)

(70)

where α _u (ω)=|ω|C _ux /(2πU) and the parameters μ _k are solutions of the following non-linear equation:

$$ 2\operatorname{cotg}(\mu_k)=\frac{\mu_k}{\alpha(\omega)}-\frac{\alpha(\omega )}{\mu_k} $$

(71)

Substituting (65), (66) into (28), (29) yields

(72)

(73)

where the sum has been truncated to consider the lowest n _w wind modes. In (72), (73) the following quantities assume the meaning of cross-modal participation coefficients:

(74)

(75)

Substituting (72), (73) in (25) yields the expression of the modal buffeting load components appearing in vector in (40).

Appendix C: Model dimension reduction

A convergence analysis of the solution of (40) with respect to the number of retained structural and wind modes (n _s and n _w , respectively) is carried out for the NCB assuming a damage intensity factor η=0.05 (5 % reduction of the whole cable section). Data are generated by the analytical model with a total length of 1-hour which, in non-dimensional time, corresponds to $\overline{t}=450$.

Figure 10(a) shows window plots of the vertical non-dimensional displacement at quarter span for a typical wind realization and considering increasing numbers of wind modes between 1 and 20 and 20 structural modes. Figure 10(b) shows a plot of the normalized relative error with respect to the solution obtained by considering 20 wind modes. The error, in Fig. 10(b), is calculated as follows:

$$ \epsilon_{n_w}=\sqrt{\int_0^1\int_0^T[\overline{v}_{n_w}(\overline {x},\overline{t})-\overline{v}_{n_{w,\mathit{max}}}(\overline{x},\overline {t})]^2\mathrm{d}\overline{x}\mathrm{d}\overline{t}} $$

(76)

where T is the total duration of the response (450 in non-dimensional time corresponding to 1 hour in dimensional time), $\overline{v}_{n_{w}}$ is the solution obtained by retaining n _w wind modes and n _w,max =20. The results presented in Fig. 10(b), normalized with respect to the maximum value of $\epsilon_{n_{w}}$ (corresponding to n _w =1), demonstrate that the solution essentially converges with 10 wind modes. Figure 10(c) shows plots of the power spectral density (PSD) of the quarter-span displacement normalized by multiplication by the frequency and by division by its maximum value. This plot shows that the first wind mode excites the first symmetric structural mode (peak 1S) but not the first antisymmetric one (peak 1A) that, instead, is activated by the second wind mode. For this reason, in Fig. 10(a), the vertical quarter-span displacement is much smaller for n _w =1 than for n _w =2. Figure 10(d) shows a plot of the envelope of the maximum vertical non-dimensional displacements along the span of the bridge. It shows that the maximum displacements are essentially well predicted already with 4 wind modes.

Fig. 10

Convergence of the analytical response with the number of retained wind modes: vertical quarter span displacement (a); normalized relative error convergence (b); quarter span normalized displacement power spectrum (norm. PSD) (c); envelope of maximum displacements along the longitudinal axis of the bridge (d)

Figure 11(a) shows plots of the vertical non-dimensional displacement at quarter span for a typical wind realization, considering increasing numbers of structural modes between 1 and 20 and considering 20 wind modes. Figure 11(b) shows a plot of the normalized relative error with respect to the solution obtained by considering 20 structural modes. The error is calculated as in (76) with n _s instead of n _w . The results presented in Fig. 11(b) show that convergence of the solution is essentially achieved by considering 10 structural modes. Figure 11(c) shows plots of the PSD of the quarter-span displacement normalized as in Fig. 11(c). Figure 11(d) shows a plot of the envelope of the maximum vertical non-dimensional displacements along the span of the bridge. It shows that the maximum displacements are essentially well predicted already with 2 structural modes.

Fig. 11

Convergence of the analytical response with the number of retained structural modes: vertical quarter span displacement (a); normalized relative error convergence (b); quarter span normalized displacement power spectrum (norm. PSD) (c); envelope of maximum displacements along the longitudinal axis of the bridge (d)