R M Sanderson and S D Smith, TWI
Paper presented at QNDE Conference 2002, Western Washington University, 14-19 July 2002
Three-dimensional finite element (FE) wave propagation models have been used to examine the effect of defect geometry and frequency on the reflection coefficients of longitudinal guided waves in 3" Schedule 40 steel pipe. An insight into the propagation characteristics of guided waves in rails is also presented. The dispersion curves for a rail are discussed and modelling results of the variation of reflection coefficient with defect depth is presented for aBS113A type rail.
Guided wave inspection systems allow pipes to be tested for corrosion with only a small section of insulation removed in an accessible region. Another possible application of the guided wave testing technique is for rails. Accidents, such as the one at Hatfield, UK in October 2000, have highlighted the need for substantial improvement of the integrity of rails, and hence also for the further development of guided wave testing techniques. Guided waves can potentially be used to save time and money in the defect detection of rails.
Finite element propagation models have been carried out in pipes and rails. Three-dimensional models have been used to study the effect of frequency and defect geometry on the reflection coefficients of guided waves in 3" schedule40 steel pipe. The variation of reflection coefficient with defect depth has also been studied for a BS113A type rail. The reflection coefficient indicates the strength of the reflected pulse from a defect and is therefore an important parameter in guided wave testing. The results provide an insight into the behaviour of guided waves in different situations and the work aids the continuing development of guided wave systems for testing pipes. The potential application of the guided waves to the large scale screening of rails is also discussed.
Guided waves offer an efficient alternative to short range techniques for screening long lengths of a structure at a time. However, the way in which guided waves propagate in a structure can be complex and, in order to reap the benefits of a fast large scale screening technique, an understanding of the characteristics of guided waves is essential.
Two modelling techniques are discussed in this paper:
- Two and three dimensional wave propagation models containing defects
- Dispersion curve calculation technique using natural frequency analyses 
The combination of these two techniques provides a powerful tool for understanding the characteristics of guided waves in prismatic structures such as pipes and rails.
The wave propagation analyses give an insight into the reflection characteristics of wave modes from defects. This is valuable knowledge at the design stage of a guided wave testing system such as for rails and also for the continuing development of an already established guided wave testing system such as Teletest ®  for pipes. The response of a wave mode to a defect is not always uniform. An example of cyclic behavior of the reflection coefficient when the axial extent of the defect is varied is presented. The natural frequency technique to find the dispersion properties of wave-guides has been in the public domain for some time.  It has relatively recently been applied to rails using finite element analysis techniques by Thompson,  Gavric  and most recently by Sanderson and Smith.  The work of Mead  , Thompson  and Gavric  was unknown to the authors at the time of the previous paper  despite an extensive literature search. The technique is useful for calculating the dispersive properties of structures of complex cross section, for which there is no analytical solution available.
The natural frequency analyses provide the displaced mode shapes of all propagating modes in a structure at any frequency. The relative amplitudes of each displacement direction through the thickness of the structure are determined by this method. This is valuable information for determining the way to excite a particular wave mode. The predicted displacements can be used to excite pure wave modes in a wave propagation model or to design a probe arrangement for a real life application.
The FE technique calculates points on the dispersion curve that can be linked up by several means. It is clearly important to link the points correctly otherwise problems in understanding the results could arise. For example, dispersion of a particular wave mode could occur unexpectedly.
Three-dimensional models were used for all of the wave propagation studies in this work. Corrosion is a common cause of flaws in pipes and is usually of a part circumferential nature. Part circumferential defects can only be modelled using three-dimensional analyses. There has been little work carried out to date using three-dimensional models probably because of the long computing times involved. In order to reduce computing time the models in this paper were assumed to have symmetry. This was possible, as the incident wave mode and the defect geometry were symmetrical in all cases.
A parametric study has been carried out on the reflection coefficient of the L(0,2) wave mode in a 3" schedule 40 steel pipe. Defect depth, and axial and circumferential length have been examined and also a range of frequencies between 40 and 80kHz have been looked at. An initial study on the effect of defect depth on reflection coefficient in a BS113A type rail has been carried out for a mode that has been found to travel purely in the head of the rail.
Parametric study of reflection coefficients in steel pipe
The finite element mesh
A symmetric half model of a 3" schedule 40 pipe was used in all of the models described in this paper. The mesh containing a 'long' defect is shown in Fig.1. The defect was modelled as 'long' in many cases so that reflections from the far wall of the defect did not interfere with reflections from the near wall of the defect. Four elements through the thickness were used. The axial length of the elements was uniform throughout the model and was taken to be a tenth of the smallest possible wavelength in the region of interest on the dispersion curve. The axial length of the elements was chosen such that there were at least 10 elements per minimum wavelength. The circumferential length of elements was chosen so that there were at least 16 elements per maximum cyclic variation around the circumference. For example, for a frequency of80kHz, the bandwidth of a 10-cycle pulse is 64-96kHz and the highest order of flexural mode in this frequency range is F(11,1). Therefore, 176 elements round the circumference were used in this example.
Fig. 1. Finite element mesh of 3" Schedule 40 steel pipe with 'long' defect
The length of all of the models used was such that the length of the incident pulse could fit into the model before hitting the first wall of the defect. The length of the model after the defect was such that the pulse transmitted through the defect would not interfere with any reflections from the defect.
Effect of defect depth
In order to study the effect of defect depth on the reflection coefficient, a mesh was created with a 'long' defect. The defect was assumed to be a rectangular notch and depths of 0, 25, 50, 75 and 100% of the wall thickness were examined. The defect had a circumferential extent equal to the radius of the pipe (~15.9% of the circumference).
The relationship between the reflection coefficient and defect depth was found to follow a shallow parabolic curve, see Fig.2.
Fig. 2. Variation of reflection coefficient with defect depth in steel pipe
Effect of frequency
Frequencies of 40, 50, 60, 70 and 80kHz were studied for a 50% wall thickness defect. As in the defect depth study the defect was a 'long' rectangular notch. The reflection coefficient was found to have little variation with frequency for this particular case, see Fig.3.
Fig. 3. Variation of reflection coefficient with frequency of excitation in steel pipe
Effect of axial extent
The defect was changed from a 'long' defect as in the previous two studies to have a finite length. The defect lengths were specified in terms of the wavelength of L(0,2). Defects of 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100% ofthe wavelength of L(0,2) were studied. As before the defect was notch like and had a circumferential extent equal to the radius of the pipe.
The reflection coefficient was found to vary cyclically with increasing axial extent. Minima were predicted for defects of 0, 50 and 100% of the wavelength of the incident mode and maxima were predicted for defects of around 25 and75% of the wavelength of the incident mode. This cyclic behavior agrees with the theoretical interaction between the reflections of the wave mode from the front and back walls of the defect. Fig.4 shows the variation of the reflection coefficient with axial extent of the defect.
Fig. 4. Variation of reflection coefficient with axial extent of defect in steel pipe
Effect of circumferential extent
For this study the defect was assumed to be crack like (i.e. zero axial length). The defect was specified in terms of the percentage of circumference of the pipe. Defects of 6.25, 12.5, 15.9, 18.75, 25, 31.25, 43.75, 56.25, 68.75,81.25 and 93.75% the circumference were studied.
The variation of the reflection coefficient with circumferential extent of the defect was found to be linear. The reflection coefficient increased with increasing circumferential extent of the defect, see Fig.5.
Fig. 5. Variation of reflection coefficient with circumferential extent in 3" Schedule 40 steel pipe
Guided waves in rails
Effect of defect depth
The dispersion curve work in the previous paper  led to a discussion of the use of modes that travel purely in the head of the rail to detect defects in the head of the rail. An initial study has been carried out on the effect of defect depth on the reflection coefficient for a mode that travels purely in the head of the rail at ~50kHz. The outline of the model is shown in Fig.6.
Fig. 6. Model of BS113A type rail containing defect with depth equal to 50% of the head of the rail
The reflection coefficient was found to have an unexpected trend with the defect depth, see Fig.7. The FFT (Fast Fourier Transform) of the reflection revealed that there were a lot of mode conversions when the incident mode hit the defect.
Fig. 7. Variation of reflection coefficient with defect depth in head of rail
The rail dispersion curve
Fig.8 shows the rail dispersion curves over the frequency range of 0-60kHz for a BS113A type rail. The three-dimensional natural frequency analysis method  was used to obtain the points for these curves and the lines were linked using an algorithm written at TWI.
Fig. 8. Rail dispersion curves up to 60kHz (dotted lines represent flexural modes and continuous lines represent torsional modes)
In order to link the points to form lines a large number of points are required especially as there are a lot of different modes in a rail at high frequencies. A two-dimensional approximation has been proposed by Wilcox et al.  for commercial FE codes. In ABAQUS  this approximation can be achieved by using axisymmetric elements that allow asymmetric deformation. In order to calculate a large number of modes using this method, a large number of Fourier variations around the circumference of the model are required. In ABAQUS the number of variations round the circumference is limited to four, as the method becomes cumbersome for more Fourier modes around the circumference.  In order to avoid this limitation the dispersion curves presented in this paper were calculated using the three-dimensional method proposed in Ref.  This also avoids any errors associated with the curved geometry inherent in the two-dimensional method.
The authors have noticed that in a recent patent application in which a rail dispersion curve was published,  the points have been linked incorrectly at around 5kHz. The lines interestingly get close at around 5kHz but do not cross one another. In the patent application the two lines are presented as crossing one another. The relationship between the two lines can be confirmed by comparing the displaced shapes of the modes for the two lines. The modes have similar displacement characteristics at this frequency; they are both symmetrical vibrations in thewhole body of the rail. However, the phase relationship between the vibrations in the head and the foot are different. For the F2 mode (see Fig.8) the vibrations of the head are out of phase with the vibrations in the foot, whereas, for the F3 mode the vibrations in the head are in phase with the vibrations in the foot of the rail. Furthermore the connectivity ofthe dispersion curves presented in Fig.8 above are in agreement with the curves presented by Gavric. 
It is clearly important to link the points correctly. In Ref.  the F2 mode has been incorrectly presented as non-dispersive around 5kHz when it is actually highly dispersive at this frequency. This would result in confusing results from field tests in which this mode were used and the predicted group velocity of the modes would have been wrong, resulting in incorrect positioning of the defect.
A deeper examination of the rail dispersion curves has revealed that there are no modes that travel purely in the head of the rail below 20kHz. At low frequencies (i.e. below ~5kHz) wave modes tend to travel in the whole body of the rail. At frequencies above 5kHz the vibrations migrate to one area of the rail for many of the wave modes. For example, the F3 mode (see Fig.8) is predominately a vertical flexing of the whole body of the rail below 25kHz, but migrates to a flexing of the head of the rail alone above 25kHz.
The wave propagation models of L(0,2) in 3" Schedule 40 steel pipe have demonstrated that there will not be a 'blind spot' in detection of crack like defects or for defects of length equal to 50 and 100% of the wavelength of the incident mode even if the defect is notch like. Moreover, real defects will be less liable to produce the same cyclic effect because they are more likely to have a continuously varying wall thickness rather than a sudden change in wall thickness.
The great number of mode conversions in the wave propagation model of the rail could be the reason why the trend of reflection coefficient with defect depth is not straightforward. The dispersion curves reveal that there is a similar mode in the rail where the energy is concentrated in the head at this frequency. The interactions between these two modes could result in a superposition that confuses the FFT result. It is intended that further work will be carried out using both wave propagation models and experiments to understand the wave modes in rails and develop guided wave rail testing equipment.
- A method for dispersion curve calculation using FE prediction of natural frequencies has provided mode shape information. This has been used to model the propagation of individual modes in rail and pipes. It also has the potential for use in the design of guided wave testing equipment.
- The reflections of L(0,2) from a rectangular defect in a 3" Schedule 40 steel pipe were studied. A cyclic variation of reflection coefficient with defect axial extent was predicted, and the reflection coefficient was nonzero even at the minimum positions of the cyclic variation. The reflection coefficient was predicted to have a linear relationship with defect circumferential extent and a slightly parabolic relationship to defect depth. The reflection coefficient was found to be independent of frequency over the range 40-80kHz.
- Reflections from a defect in the head of a steel rail have been analysed. For a wave that propagates purely in the head of the rail, the reflection coefficient was not clearly related to the defect depth.
The authors would like to thank the members of TWI for providing the funding to make this work possible.
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