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Modelling guided waves in complex structures - Part 1: Rail


Yousef Gharaibeh1,2, Chiraz Ennaceur1, Peter Mudge1, Wamadeva Balachandran2

1Long Range Ultrasonic (LRU) Section, TWI, Cambridge, UK

2School of Engineering and Design, Brunel University, Uxbridge, UK

Paper presented at NDT 2009, Blackpool, UK, 15-17 Sept. 2009.


Long Range Ultrasonic Testing (LRUT) uses guided waves in the kilohertz range to inspect many metres of an elongated component from a single point of access. This technique is well developed for structures having a simple geometry, such as plates, rods and pipes. However, this is a relatively new technology and there is still much to learn about the behaviour of guided waves and their application in complex structures. Mathematical modelling techniques provide an efficient method of investigating this behaviour. This paper illustrates how modelling techniques have been used to design long range tests for railway rails. Part 2 of this work concentrates on application of modelling to wave propagation in plastic coated wire bundles.

Utilising ultrasonic guided waves for the inspection of rails is challenging. This is mainly due to the irregular cross-sectional geometry of the rail, the existence of a large number of wave modes in the kilohertz range and the presence of a substantial dispersion effect. The aims of this work were:

- To identify suitable wave modes that can be generated and propagated in each section of the rail (i.e. the head, web and foot),
- To find a suitable means of excitation (i.e. transducer design) for the selected wave modes
- To demonstrate the ability of the identified wave modes to detect flaw

ABAQUS CAE, a Finite Element Analysis (FEA) software package, was used as a tool to provide a better understanding of the behaviour of guided waves in the rail structure. The FEA tool was used to examine the possibilities for generating and propagating guided waves in rails. Experimental trials were also conducted to validate the findings of the FEA. The findings demonstrate the ability of the identified wave modes to detect a range of different types of flaw at different locations within the rail cross-section over long test lengths.

1. Introduction

Inspecting rails for defects is a major concern for the rail network industry. In Europe the economic cost of rail failure is around €2 billion per year.[1-2] Rail failure can cause train derailment which might lead to a catastrophic loss of human lives and commercial loss. The most prevalent failure mechanism is the transverse fatigue crack.[1-2] 39.5% of rail failures in the UK rail network are due to defects in the transverse plane of the rail, while 22.4% are caused by defects in the alumino-thermic welds.[3] The Hatfield (UK) rail accident is an example of a deep growth of transverse defect in the rail head. This incident caused the death of 7 passengers and serious injuries to a further 11.[2-3] The rail web and foot are also subject to cyclic stresses and may suffer fatigue cracking and corrosion as a consequence of operational and environmental conditions. [4-6]

Guided-waves offer significant advantages over other inspection methods for the economically efficient inspection of rails as the long propagation distances allow considerable lengths to be examined from a limited number of test locations and they provide 100% coverage of the cross sectional area of the rail[3]. However, the drawback to the use of guided waves in rails is that the wave modes which exist are more complex those in plates or pipes. This makes utilisation of guided waves in rails extremely challenging. Finite Element Analysis (FEA) was used as a tool to give a better understanding of the behaviour of guided waves in the structure in terms of identifying suitable wave modes that can be excited and propagated in each section of the rail. This allows prediction of optimum mode excitation, and demonstration of the ability of the selected wave mode to detect different types of flaw. Experimental trials were conducted to validate the findings of the FEA. This paper presents the findings of both the modelling and the experimental work.

2. Application of Finite Element modelling to predict the behaviour of guided waves in complex structure

Finite Element modelling has been used successfully by a number of workers for predicating and simulating the behaviour of guided waves in complex structures such as rails[7-11]. For this work the ABAQUS Explicit FEA package was used, which uses the central difference rule to integrate equations of motion explicitly through time at each increment. In each case a tone-burst pressure pulse was applied to represent the ultrasonic pulse and displacement vibrations of the reflected signals from the section modelled were monitored at the points of excitation, i.e. the models represented pulse-echo tests.

The study was carried out on a representation of a rail section, according to BS113A[12], with a length of 6m, a density of 7850kg/m3, Young's modulus of 207GPa, and Poisson's ratio of 0.33. A ten cycle tone burst signal was applied as a load to the model. The element length was equal to a tenth of the minimum wavelength. This enabled the model to simulate the wave modes present efficiently. The element type used in the models was an 8-noded reduced integration brick (ABAQUS element type C3D8R).

2.1 Mode identifications

Knowledge of the dispersion curves for the potential wave modes is very important as they describe wave velocity at a given frequency. In a rail there are a large number of possible wave modes due to the rail's complex geometry. A number of authors have derived dispersion curves for rails using different numerical techniques such as Semi-Analytical Finite Element Model (SAFEM), FEA, and analytical forms.[9-11] In this paper, dispersion curves are used from a previous work carried out by Sanderson.[10-11] Figure 1 shows dispersion curves for a rail conforming to BS113A. From these, the large number of possible wave modes is evident. Further, the phase velocity of these varies widely with frequency.

Fig. 1. Rail dispersion curves BS113A; for all possible wave modes between 0-60kHz
Fig. 1. Rail dispersion curves BS113A; for all possible wave modes between 0-60kHz

For this work the rail cross section was divided into three sections; head, web and foot (see Figure 2). The aim of this study was to identify one wave mode in each section, which would be used for the inspection, and to generate only the selected mode. This was to allow each section to be examined individually, to avoid any interference from the other parts of the rail and make the data readily interpretable. The three wave modes identified for study are described below.

Fig. 2. Rail geometry BS113A
Fig. 2. Rail geometry BS113A

As Figure 3 shows the selected deformation shape of each wave mode at each section of the rail. [10-11]

  • Head: The so-called F3 mode[10-11], which is a vertical flexing mode (Figure 3a)
  • Web: The so-called T2 mode[10-11], which is a horizontal flexing mode (Figure 3b)
  • Foot: The so-called F2 mode[10-11], which is a vertical flexing mode in the foot (Figure 3c)
Fig. 3. Different wave modes for different section in the rail
Fig. 3. Different wave modes for different section in the rail

a) F3 wave mode in the head
b) T2 wave mode in the web
c) F2 wave mode in the foot

These modes have been chosen for the following reasons:

  • Each mode exists solely in one part of the rail with little expected leakage into the other parts,
  • The displacement of each mode exists in the full section thickness of the respective part of the rail, suggesting that 100% coverage of the rail cross-section may be achieved,
  • The pattern of displacement is similar for all three modes, so that similar transducer arrangements could be used for their generation and reception,
  • They are all relatively non-dispersive in the 20-80 kHz range, so that their propagation characteristics are favourable for defect detection. They also have a similar group velocities. Figure 4 shows a sub set of the dispersion curves, as presented in Figure 1, for these modes.
Fig. 4. Rail dispersion curves BS113A for selected waves
Fig. 4. Rail dispersion curves BS113A for selected waves

2.2 Propagation of modes F3, T2 and F2

Based on the displacement characteristics of each mode, the appropriate excitation pattern was applied to the model in each case in order to generate the specific modes. Figures 5-7 show the propagation of the selected wave modes along the rail section modelled. The velocities derived from the models are given in Table 1.

Fig. 5. Modelling results for F3 propagated in the Rail head
Fig. 5. Modelling results for F3 propagated in the Rail head
Fig. 6. Modelling results for T2 propagated in the Rail web
Fig. 6. Modelling results for T2 propagated in the Rail web
Fig. 7. Modelling results for F2 propagated in the Rail foot
Fig. 7. Modelling results for F2 propagated in the Rail foot

Table 1 Wave velocities derived from the propagation model

Selected Wave mode Phase Velocity (m/s) Group Velocity (m/s) F3 2830 3100 T2 2584 3000 F2 2614 3027

These results show that:

- By applying the appropriate excitation signal to each part of the rail the desired wave modes can be successfully generated,
- Whilst the T2 mode in the web shows little leakage of energy into the other parts of the rail section, the results for both the F3 mode in the head and the F2 mode in the foot show that there is leakage of ultrasonic energy into the other parts of the rail. It was considered necessary to investigate experimentally whether this leakage would affect the ability to perform a satisfactory inspection.

Figures 8-10 show the pulse-echo responses from the 6m long model at the excitation point (i.e. the simulated transducer location), without any defects present. For these the time has been converted to distance using the above velocities to show the position of the reflectors relative to the excitation point. NB. The distance axes show distance travelled by the ultrasonic pulse, not range. These results were obtained at 70kHz.

Fig. 8. FEA generated response for F3 in the Rail head
Fig. 8. FEA generated response for F3 in the Rail head
Fig. 9. FEA generated response for T2 in the Rail web
Fig. 9. FEA generated response for T2 in the Rail web
Fig. 10. FEA generated response for F2 in the Rail foot
Fig. 10. FEA generated response for F2 in the Rail foot

These results show that:

- In each case the response from the far end of the 6m section modelled may be clearly seen,
- Velocity: the travelling velocities (the group velocity) for these wave modes are approximately as predicted by the models, but are not exactly as expected. This difference may be accounted for by the fact that these waves are not truly non-dispersive (see Figure 4). This can cause the leading and lagging edges of the tone burst signal to travel at two different velocities; a consequently the travelling wave mode becomes elongated as it travels in the structure. Hence; the observed variations between the group velocity and phase variations.
- Mode conversion: The signals represented in the above Figures show the response after reflection from the far end of the model. Owing to the large number of wave modes possible (see Figure 1) a significant amount of mode conversion is likely. The interaction of the incident signal with the rail end causes such mode conversions, as observed on the traces.
- Excitation conditions: The point source excitation exhibits constructive and destructive interference to generate the shaped pulse. This interference is identified as ringing effect. The ringing effect is coupled with complex mode conversion for the wave, as excitation points source was located near the rail end. Hence, the high ringing effect exhibited at the beginning in the response signal.
- Leakage: Figures 5-7 show that some leakage of energy into other wave modes will occur. These will travel at different group velocity from the desired mode and may be dispersive, leading to unwanted signals on the output.

3. Experimental validation

In order to validate the FEA findings and to establish their credibility to predict wave propagation in a variety of circumstances it is important to carry out experimental verification of the models. For this purpose a 6.64 m long rail specimen to BS113A was used. This is shown in Figure 11. The specimen had two discontinuities in the web. One was a stud-welded aluminium wire attachment and the other was a through hole with 30mm diameter. The pulse-echo transducers were mounted at one end of the specimen, while the discontinuities were towards the far end. They were located at 6m and 6.22m from the transducer respectively.

Fig. 11. The 6.64m rail specimen with two discontinuities in the rail web, as shown
Fig. 11. The 6.64m rail specimen with two discontinuities in the rail web, as shown

3.1 Generation of the selected identified wave modes (F3, T2 andF2) and flaw detection

The focus of this work was to generate the selected wave modes (F3, T2 and F2) in the rail specimen experimentally; based on the parameters derived from FEA as described in section 2. For the purposes of the experiment, piezo-electric elements were bonded on to the rail surface to generate guided waves in the rail. These were placed one wavelength from the end of the rail. A Plant Integrity Teletest® unit was used as an arbitrary function generator to excite the transducers and to collect the responses from the rail.

Figures 12-14 show the pulse-echo response of each wave mode in the different sections of the rail. The wave is reflected from the rail end at a distance of approximately 6.7m. However, due to the additional features in the web; three reflections were recorded for T2, see Figure 13. This result shows the ability of T2 to differentiate between different features within the same section. These reflections were recorded at 5.98m, 6.4m and 6.7m respectively. The first two reflections are an indication to the two discontinuities, while the last reflection is from the rail end.

From these experiments it was clear that:

- The desired wave modes could be generated successfully in the three different parts of the rail,
- The T2 mode was sensitive to the two discontinuities present in the web,
- The modes propagated in the head and foot did not show any responses from these discontinuities, indicating that the different modes are well confined to the appropriate parts of the rail and the level of leakage does not affect test result.

Fig. 12. Experimental result for F3 propagating in the Head
Fig. 12. Experimental result for F3 propagating in the Head
Fig. 13. Experimental result for T2 propagating in the web
Fig. 13. Experimental result for T2 propagating in the web
Fig. 14. Experimental result for T2 propagating in the foot
Fig. 14. Experimental result for T2 propagating in the foot

Finally, the results from the experimental trials are behaving in the same manner as the FEA results. Both the ringing effect and mode conversion appear to be similar in nature to the model prediction.

In addition, there is this variation in the measured velocity, hence the variation in the recorded distance. This is can be as mentioned previously due to the natural complexity of guided waves and experimental conditions. Experimental conditions can include; contact conditions between the piezo-electric elements and the test piece, the spacing precision between the elements are also important factors that can affect the propagation of guided waves, the excitation location and the purity of the steel.

4. Discussion

The transient wave mode models for each of the three sections of the rail (Figures 5-7) showed that the waves are travelling in the same displacement distribution pattern as described theoretically modal vibrations in Figure 3. This predicted that the rail could be examined in three distinct parts using the selected wave modes.

The results from the experimental tests showed that there was good correlation between the FEA prediction of the responses from the rail specimen and those from the practical tests, thereby demonstrating that this model could be used to design test procedures for rails. In particular, the experimental results showed that:

- The wave modes selected in the modelling exercise could be propagated in each part of the rail, in the manner predicted by the model.
- The transducer arrangement for each mode, designed on the basis of the input displacement pattern derived in the model, was shown to be successful in generating the chosen mode in each case.

FEA also predicts that there is a wave mode leakage to other sections of the rail for all three modes studied. Also, FEA predicts that there is a substantial presence of the dispersion effect, mode conversion, ringing effect, and un-wanted 'leaky' wave modes. Their level of significance can have an impact on the signal interpretations. These effects were also observed in the practical test results, again indicating that the model represents the practical case well and also indicating that there may be limitations of performance of the practical tests owing to these factors. Despite these, the test on the web using the T2 mode managed to discriminate between different web rail features and the far end of the rail.

5. Conclusion

The behaviour of guided waves in rails is complicated. This is due to the irregular cross section of the rail, which results in irregular mode of vibration in comparison with simple cross section such as pipes and plates. This results in difficulties in generating and propagating specific wave modes. In addition, many of the guided waves are dispersive. This makes utilising guided waves in rail challenging. However, this research has demonstrated:

- That specific wave modes can be identified, which may be used for testing different parts of the rail.
- The ability of the identified wave modes to be generated and propagated in the rail for many metres (up to 12m) from one access point.
- The ability to detect discontinuities in the web.

However, there is a need for further study of the detection of different types of defects in other sections in the rail and to determine the sensitivity of these tests.


Yousef Gharaibeh is studying for an Engineering Doctorate at Brunel University under the sponsorship of EPSRC and TWI Ltd. Some of the work was carried out under the Long Range Ultrasonic Condition Monitoring project, which was partly funded by the European Commission under the initiative: 'Horizontal Research Activities involving SMEs' within the Framework 6 Programme, contract Coll-CT-2005-516406.

Part 2


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