# Modal analysis of road and rail bridges for damage detection and resonance prediction

Lutz Auersch, Samir Said, Rolf Rohrmann

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Lutz Auersch, Samir Said, Rolf Rohrmann. Modal analysis of road and rail bridges for damage detection and resonance prediction. 11th International Operational Modal Analysis Conference, Institut National de Recherche en Informatique et en Automatique [Inria], May 2025, Rennes, France. hal-05398806

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# Modal analysis of road and rail bridges for damage detection and resonance prediction

Lutz Auersch¹, Samir Said, and Rolf Rohrmann

¹ Federal Institute of Material Research and Testing (BAM), Berlin, Germany, lutz.auersch-saworski@bam.de

## ABSTRACT

In the 1980s, the Federal Institute of Material Research and Testing started with modal analysis measurements of some bridges before and after repair. For one of the bridges, a structural health monitoring was installed 1994 which is still working up to now. It has been modified and extended several times. The monitoring was extended from the critical span to three neighbouring spans. A modal analysis of the whole bridge with seven spans have been done three times, twice together with EMPA of Switzerland. Additional calibration measurements have been done and additional evaluation procedures have been implemented for the monitoring of the steadily increasing loads from the road traffic. Additional sensors were installed such as strain gauges, crack-width, and temperature sensors. The strong influence of the temperature on the natural frequencies has been studied over the years. Later, a temperature compensation has been established and a weak aging trend has been found in the monitoring data. Now, the bridge will be demolished and replaced by a new bridge. Some results of this long-term monitoring will be shown and possible damages (changes of the pre-stress or the support structure) will be discussed. A second application of modal analysis will be demonstrated: the prediction of the resonances due to passing trains. The response of a bridge to passing trains can be calculated in frequency domain as the multiplication of three spectra, the axle sequence spectrum of the train, the transfer function of the bridge, and the modal force spectrum of a single passing load. A resonance occurs if a maximum of the train spectrum coincides with the maximum of the bridge spectrum. The amplitude at this resonance is strongly influenced by the modal force spectrum which is identical to the frequency or wavenumber spectrum of the corresponding mode shape. Therefore, modal analysis from calculation, impact measurements, wind and train measurements are necessary for the prediction of the resonance occurrence and amplification. Examples of mode shape spectra for single or multi-span bridges with simply supported or continuous spans will be shown, and some relations between mode shapes and resonance amplifications will be concluded.

Keywords: Bridge monitoring, Multi-span bridges, Damage detection, Resonance
# 1. INTRODUCTION

In the 1980s, the Federal Institute of Material Research and Testing (BAM) started with the research on bridges and the damage detection with dynamic methods [1]. For one of the bridges, a structural health monitoring was installed 1994 which is still working up to now. The modal analyses of this 7-span highway bridge will be discussed in Section 2, as well as experiments on a damaged 2-span beam in Section 3, and the resonance of multi-span railway bridges in Section 4.

# 2. MONITORING AND MODAL ANALYSIS AT THE WESTEND-HIGHWAY BRIDGE

The Westend highway bridge has been built in the early 1960s as a pre-stressed concrete box girder bridge. It has a length of 240 m and consists of seven continuous spans of approximately 35 m length and 14 m width (Fig. 1a) which are supported on hinged circular columns, the abutments at the end, and a clamped column in the middle of the bridge.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** The 7-span Westend bridge (a), averaged spectrum (b), mode shapes of the third span from measurement (c,d) and finite-element calculation (e,f), second bending (c,e) and second torsion (d,f).</figcaption></figure>

In the late 1980s, the dynamics group of BAM started with modal analysis measurements of some bridges before and after repair [2] including the Westend bridge. The preliminary studies concentrated on the critical third span where cracks in the box girder and in the shear wall above the column occurred. The modal analysis of this span clearly showed the first and second bending and torsion modes between 2 and 9 Hz (Fig. 1c-f). The first bending mode appears several times with different frequencies. The effect of different excitations, for example microseismic, impacts, sudden release, near-by rail traffic, light and heavy road traffic had been analysed and a first beam model of the whole bridge was updated [3].

A monitoring system was installed and started 1994 [4] which is still working up to now. It has been modified and extended several times. It was extended from 16 to 32 and 48 channels, from the critical span 3 to the neighbouring spans 1, 2 and 3. Geophones (velocity transducers) are used for the vibrations, eigen frequencies and mode shapes. Acceleration sensors and a seismometer have been used to check the accuracy at the low-frequency eigen frequencies. The monitoring software includes the measurement and the immediate evaluation of the raw data [5]. The characteristics of the vibration such as max/min values, root mean square values, frequency peaks are extracted and stored together with selected time segments, single and averaged spectra. Additional sensors were installed such as
temperature, strain gauges, inclination, and crack-width sensors. A calibration measurement with a 60 t truck has been done and additional evaluation procedures have been implemented for the monitoring of the steadily increasing loads from the road traffic.

Three times, modal analysis measurement campaigns have been performed on the whole 7-span bridge, twice together with EMPA of Switzerland [6]. The spectrum shows several clearly separated peaks (Fig. 1b). The mode shapes in Figure 2 show coupled bending of the different spans where all spans have alternating minima and maxima at 2.8 Hz, pairs of neighbouring spans are in phase at 3.5 Hz, and all spans are in phase at 4.5 Hz. A torsional mode with alternating minima and maxima is found at 10.4 Hz. Note, that the three modes at 2.8, 3.5 and 4.5 Hz give almost the same fundamental mode shape at span 3 which has often the highest amplitude of all spans. The different amplitudes of the different spans can be due to the differences in the span lengths of 31 to 38 m, the varying curvature and inclination of the bridge, or the clamped support between spans 3 and 4.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Selected mode shapes of the whole 7-span bridge from measurement.</figcaption></figure>

The stored monitoring data have been analysed for the influence of the traffic loads [7] and the temperature [8]. The monitoring data of the years 2000 to 2014 have been further evaluated by Hu in the context of statistical pattern recognition [9] (Fig. 3). The influence of temperature has been removed and a trend of some health indices has been found which has been related to a loss of prestress.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Monitoring data of the Westend bridge, a) eigenfrequencies, b) strain of a tendon.</figcaption></figure>
### 3. MODAL ANALYSIS AND DAMAGE DETECTION AT A 2-SPAN MODEL STRUCTURE

A two-span pre-stressed concrete beam with a total length of 24 m (2 x 12 m) and a width of 1 m has been built on the BAM test area (Fig. 4a). A line of 25 vertical geophones were installed at 1 m intervals along the bridge, and the modal analysis has been done by hammer tests. The natural frequencies appear in pairs, where the square roots of the frequencies always keep the same distance (Fig. 4b). The first four modes are presented in Figure 4c-f as theoretical modes (lines) and experimental results (square symbols). Each pair of natural frequencies belongs to the antimetric mode with a lower frequency (Fig. 4c,e) and the corresponding symmetric mode with a higher frequency (Fig. 4d,f). The agreement between theory and experiment is very good.

The beam has been damaged by releasing the prestress, and many cracks developed. A second modal analysis was done in this released state, and then the beam was prestressed again and a third measurement was done. A clear reduction of the first eigenfrequencies has been found for the intermediate state without prestress and with subsequent damage, but the eigenfrequencies are almost identical for the undamaged and damage state with prestress (Fig. 4g). That concludes that cracks under prestress are difficult to detect by modal analysis. This could also be concluded from the monitoring of the Westend bridge where cracks in the bridge span appeared. Moreover, three of 38 tendons had been accidentally cut on the one side of the span. No clear effect could be observed as the prestress is still closing the cracks. If there is a degrading trend in the monitoring data, a damage in the cracked shear walls above the columns with minor reinforcement is more likely. Such an observation, that damage at the support has stronger effects than damage at the tendons, has been made at the three-span bridge S101 in [10].

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Two-span beam (a), regular sequence of the natural frequencies (theory) and ☐ measurements (b). Calculated mode shapes of the two-span beam of length 2l, first, and second antimetric (c,e) and symmetric (d,f) mode, ☐ measurement results.</figcaption></figure>

### 4. RESONANCE OF RAILWAY BRIDGES

A 45 m long, single-span concrete box-girder bridge has been measured under the passage of a test train with different speeds [11]. A modal analysis was performed from the vibrations after the train passage. The first bending mode was found at 3.7 Hz, and the first torsion mode was at 6.7 Hz. The second bending mode followed at 11.2 Hz, and the second torsion mode was at 12.7 Hz. Train passages with 100 km/h excited only one mode, the first torsion mode, whereas the train passage with 160 km/h excited only the second bending mode (Fig. 5a). The preference for a certain mode is not so surprising, but the absence of the other modes was.
The passage of a train excites the different modes of the rail bridge with (angular) eigenfrequencies $\omega_j$ and mode shapes $w_j(x)$. The solution for the bridge vibration can be expressed in frequency domain as the superposition of the eigenmodes where each contribution is a product of three spectra [11]

$$v_j(\omega) = F_s X(\omega) W_j(\omega) H_j(\omega) \quad (1)$$

where $F_s$ is one static axle load, $X(\omega)$ is the axle-sequence spectrum, $W_j(\omega)$ is the spectrum of the mode shape $w_j(x)$, and $H_j(\omega)$ is the transfer function of the bridge. It holds for any type of rail bridges, simply supported, elastically supported, integral, single-span, multi-span, and continuous bridges. The time domain solution is obtained via the inverse Fourier transform.

The modal force for a moving constant load $F_s$ is

$$F_j(t) = \int F_s \delta(x - v_T t) w_j(x) dx = F_s w_j(v_T t) \quad F_j(\omega) = F_s W_j(\omega) \quad (2)$$

in time and in frequency domain where $v_T$ is the train speed. The spectrum $W_j(\omega)$ is the Fourier transform of the mode shape $w_j(v_T t)$.

The frequency response function $H_j(\omega)$ for the velocity is

$$\frac{v_j(\omega)}{F_j(\omega)} = H_j(\omega) = \frac{i\omega}{m_j(\omega_j^2+2D_j i\omega_j^2-\omega^2)} \quad (3)$$

and its maximum at the resonance frequency $\omega_j$ is

<figure><img src="image_6.png" /><figcaption>**Figure 5.** The passage of the test train over the 45 m long bridge with 100 km/h (left) and 160 km/h (right), time histories and spectra, a) measurement, b) modal force and c) transfer function of the first torsion and the second bending mode, d) axle-sequence spectrum e) bridge response calculated with the first four modes.</figcaption></figure>
$$ \frac{v_j(\omega_j)}{F_j(\omega_j)} = H_j(\omega_j) = \frac{1}{m_j 2 D_j \omega_j} \qquad (4) $$

where $D_j$ is the modal damping, and $m_j$ is the modal mass

$$ m_j = \int m' w_j(x)^2 dx \qquad (5) $$

The resonance amplification is inversely proportional to the modal mass, the modal damping and the (angular) eigenfrequency.

This frequency-domain method is demonstrated for the example railway bridge in Figure 5b-e. The modal force spectrum (Fig. 5b) is an impulse spectrum with many periodical zeros defined by the passage time $T$. The envelope of the spectrum is decreasing with frequency. The transfer functions of the railway bridge for the two measured natural frequencies have nearly the same the maximum amplitudes (Fig. 5c). The test train consists of two locomotives and five passenger cars. The axle-sequence spectrum for 100 km/h (Fig. 5d) has a first minimum region around 5 Hz, a maximum region around 7 Hz and a second minimum region around 12 Hz for 100 km/h. This explains the dominance of the torsional mode at 6.7 Hz and the cancellation of the other modes for 100 km/h. These minimum and maximum regions are shifted for 160 km/h (Fig. 5e) and the torsional mode is now cancelled whereas the second bending mode is amplified. The axle sequence spectrum of the train can well explain the measured effects of Figure 5a.

The effect of multi-span bridges is discussed for a high-speed train on a 15 m short bridge. A resonance at the first bending eigen frequency at 10 Hz can be excited by a car-length peak of the train, for example by the third car-length frequency of the ICE3 with 280 km/h [11]. Figure 6 shows the modal force spectra for a single, for two, and for four identical simply supported bridges. For the single-span bridge (Fig. 6a), the second maximum of the modal force spectrum is at the resonance frequency of 10 Hz. Two bridge spans have two coupled modes at close frequencies, the first is in anti phase, the second is in phase (Fig. 6b). Four fundamental modes exist for the four-span bridge (Fig. 6c,d). The first fundamental mode (Fig. 6c) with alternating maxima and minima has only limited spectral amplitudes

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Modal force in time and frequency domain a) for a 15 m long single-span railway bridge, 280 km/h train speed, b) for the out-of-phase and the in-phase fundamental mode of a two-span bridge, c,d) for the four fundamental modes of a four-span bridge.</figcaption></figure>
around 10 Hz and may even have a zero at 10 Hz. The intermediate fundamental modes (Fig. 6c<sub>2</sub>,d<sub>1</sub>) with some sign changes have similar effects with a little higher spectral maximum. Only the highest fundamental eigenfrequency (Fig.6d<sub>2</sub>) with its in-phase mode shape has a maximum at 10 Hz which is proportional to the number *n* of spans. The increase of the modal force maximum must be compared with the increase of the modal mass, which reduces the transfer function of the bridge and which is proportional to *n*. Therefore, the first fundamental eigenfrequencies have lower response amplitudes, and only for the highest fundamental mode, the effects of the modal force (the mode shape) and the modal mass compensate and the resonance amplitude is the same as for the single span bridge.

Similar rules hold for continuous bridges with *n* equal spans, but with little different mode shapes and different eigen frequencies which are separated and distributed between the simply-simply supported eigenfrequency *f*₁ and the clamped-clamped eigenfrequency 2.25 *f*₁ of a single-span bridge. The first fundamental modes have lower resonance amplitudes, and only the highest fundamental frequency has similar high amplitudes as a single clamped bridge span. The four fundamental mode shapes measured at a continuous 230 m long four-span bridge from [12] are shown in Figure 7, and they are as regular with equal min- and max amplitudes as the calculated mode shapes in Figure 6c,d.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** A four-span highway bridge (a) and the four fundamental mode shapes from measurement (b-e).</figcaption></figure>

So far, only mode shapes with equal amplitudes for all simply supported spans have been considered. In case of different amplitudes for different spans (as measured for the 7-span Westend bridge, or for the three-span rail bridge in [11]), a closer approach to the single-span bridge can be expected. Certain amplitude combinations can be found which result in even higher resonance amplifications. This theoretical upper limit, however, will rarely occur, and a reduced resonance amplitude is more probable. The special behaviour of multi-span bridges has also been discussed for the structural health monitoring [13].

**5. CONCLUSIONS**

A seven-span highway bridge has been monitored during 30 years for damage detection. A weak degrading trend has been observed. Several modal analyses have been performed for a single span and for the whole bridge. A second task, the resonance of railway bridges and its dependence on the mode shape of multi-span bridges, has been demonstrated and may attract other researchers on modal analysis.

**ACKNOWLEDGEMENTS**

Thanks to W. Schmid for providing the sensors and the monitoring system, to the whole dynamics group for the help in the measurements, and to its head W. Rücker.
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