Lamb Waves in Plate Girder Geometries

D.W. Greve,1 N. L. Tyson2, and I.J. Oppenheim2

1 Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213 2 Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213

ABSTRACT Lamb waves generated by wafer-type transducers have been previously studied in rectangular plates, and in this paper we describe finite element simulations and laboratory experiments extending those studies to plate girders. A Lamb wave generated in the web of a girder is shown to induce a wave traveling at a different speed in the flanges, and to develop a wavefront geometry in the web that is influenced by reflection from the web-to-flange joints. We compare simulation results to waves measured on steel plate girder specimens. The topic is of interest because steel plate girders must be inspected or monitored to detect fatigue cracks, and this work provides information on the area that may be illuminated by each wafer-type transducer, coupled to the orientation and size of fatigue crack that might be resolved when Lamb waves are used for pulse-echo flaw detection.

I. Introduction Steel plate girders are the most common highway or railway bridge structural member and are very efficient for their purposes; however, they incur fatigue cracks and must be inspected frequently. Presently inspection is a visual mapping process in which crack location, orientation, and length are recorded. We seek to develop resident active sensors to monitor cracks at critical locations; we believe that such a technology would improve structural health monitoring practices, and we note that there would be particular benefits for the inspection of cracks in locations that are otherwise difficult to access. We have previously conducted experimental and simulation studies of Lamb waves generated in thin elastic plates by piezoceramic wafer-type transducers [1]. Those studies confirmed our expectations that Lamb waves can return reflections from cracks or other discontinuities, and produced simple guidelines for choosing the transducer dimensions and the forcing function for selective mode generation. Because plate girders are fabricated from thin plates, we expect Lamb waves to be useful for monitoring their structural condition. However, the many welded joints and complex shapes make it difficult to interpret reflected waveforms. In this paper we combine finite element simulations with selected experiments to understand some aspects of wave propagation in these structures. Simulations were performed with FEMLAB 2.3 or FEMLAB 3.0 (in two and three dimensions, respectively) in the time-stepping mode. The excitation was a 5-cycle pulse, a shaped sinusoid with center frequency f, of the form

⎩⎨⎧

≥<

=ftftftft

tF/50/5)sin()2sin(

)( 102 ππ .

Details of the approach used for simulation in two dimensions are published elsewhere [1]; in three dimensions the GMRES linear system solver and the weak solution form were used. Solution output times were chosen to be roughly 1/8th the period of the center frequency, and the element size was generally chosen to be a fraction of the wavelength. The computational burden of this latter condition, as reflected in the number of elements used in the simulation, is much more modest in two dimensions than in three dimensions. For that reason, simulations for three-dimensional problems were limited to smaller regions and required considerably greater computation time.

II. Simulation of reflections from a part-thickness crack in a plate Reflections from flaws in a plate have been simulated by others including Rose and coworkers [2] and Lowe and Diligent [3]. We report here simulations of Lamb wave generation by a wafer-type transducer, and the interaction of the generated wave with a part-thickness slot. Simulations were performed for an aluminum plate 1.59 mm in thickness with a piezoelectric emitting transducer 0.64 cm long and 0.064 cm thick, containing a rectangular slot 0.01 cm wide penetrating halfway through the plate thickness. The simulations were performed for a plate 0.6 m in length, symmetric about x = 0, excited by a pulsed voltage waveform with center frequency f = 400 kHz.

Figure 1. Particle velocity (y-direction) at bottom surface

The analysis was performed under the assumption of plane strain, reducing the problem to two dimensions. Figure 1 plots the y velocity at the bottom surface as a function of time. It is convenient to plot the y velocity because A0 and S0 modes both have significant displacement normal to the surface. Consequently both modes can be seen in the same plot. At t = 25 µs we see both A0 and S0 modes to have been generated by the transducer. With wafer transducers, both modes are generated except in special cases where the mode wavelength is an integral multiple of the transducer length [1],[4]. At the center frequency of 400 kHz, the A0 mode is slower, has a shorter wavelength, and exhibits more dispersion than the S0 mode. The S0 mode begins to interact with the slot at t = 35 µs and the A0 mode begins to interact with the slot at t = 55 µs. Interaction of the S0 mode with the slot results in four distinct pulses: transmitted and reflected S0 modes, and transmitted and reflected A0 modes produced by mode conversion. The A0 mode also results in transmitted and reflected pulses, although mode conversion to S0 is not discernible in these simulation results.

The large number of distinct pulses resulting from interaction with simple defects represents a challenge for the practical application of Lamb waves for flaw detection. In general it will not be practical simply to look for defect-related echoes as is done in traditional ultrasonic inspection. Options include: various techniques for comparing waveforms [5]; careful design of the transducer to reduce the number of emitted modes [4]; and time-reversal acoustics [6]. In all cases it is useful to understand the interaction of various types of defects with Lamb wave, especially in complex geometries. In the remainder of this paper we consider three different specific cases of particular interest.

III. Simulation of reflections from a crack at a web-flange welded joint We studied reflections occurring at a welded joint between a web 0.63 cm thick and a flange of varying thickness. Figure 2 shows both the geometry of the joint, together with a model of a crack halfway through the web thickness at the weld location.

Figure 2. Reflection of an incident S0 wave at a welded joint with a crack

We simulated the reflections of an S0 mode, originating in the web, with and without the crack. Simulations were performed with a center frequency of 200 kHz. Figure 2 shows the simulation results for a flange thickness of 1.26

cm (in Figure 2 the girder is rotated 90o from its customary orientation). In this plot the arrows indicate the displacement, and colors indicate the von Mises stress, with red corresponding to the highest stress. We see the S0 wave approaching the joint at 50 µs. At 60 and 70 µs the wave begins to interact with the joint. At 80 and 90 µs there is a reflected S0 wave leaving the joint, together with waves propagating into the flange. In order to quantify these results we have calculated the total energy in the reflected S0 mode relative to the energy in the incident pulse. The results are shown in Fig. 3 as a function of flange thickness, with and without the crack present. Without a crack the reflected energy is small, particularly when the flange is thicker than the plate, which is the usual situation in a structural girder. The reflected energy is seen to reach a minimum for a flange thickness of 1.90 cm, or 0.75 in. At this thickness there is a good match between the displacements of the incident S0 mode and an S1 mode in the flange (not shown). When the crack is present, the reflected energy is substantially increased, indicating that crack detection should be feasible.

Figure 3. Reflected energy from welded joint with and without the presence of a crack for various flange thicknesses.

However, in structures we must consider reflections in a complex three dimensional geometry. Simulations in this case are more difficult because of the large number of elements that are needed to model the waves. In the following section we present some simulations of small regions that provide insight into this difficult problem.

IV. Simulation of wave propagation in a rolled beam

Figure 4. Displacement in the z direction for a wave excited in a rolled beam

We next consider propagation of guided waves originating at mid-height of the web in a rolled beam. Figure 4 shows the propagation of 150 kHz waves in a beam 15 cm deep, with flanges 7.5 cm wide and a web 0.5 cm thick; the beam is again rotated 90o from is customary orientation, with planes of symmetry about z = 0 and x = 0. An S0 mode was generated by applying a time-dependent force in the z direction to an area element at the origin; this is roughly equivalent to a point-force excitation in the two dimensional problem. In Figure 4 the colors indicate displacement in the z-direction, with red positive and blue negative. The wave is initially launched with a semicircular wavefront that becomes nearly a straight wavefront as propagation proceeds. We also see a wave reflected from the flange, which lengthens the tail of the pulse propagating in the web, as well as some propagation into the flange. Figure 5 shows the z displacement as a function of time for various locations along the centerline of the beam; the tail extending well past the duration of the exciting pulse is clearly visible.

Figure 5. Simulated z displacement as a function of time at various distances along the axis of the beam.

V. Simulation of wave propagation in a plate girder Rolled beam products are limited in overall depth and in their proportions of width to thickness. Most steel bridges are instead designed as plate girders, in which thin plates are welded together to create an optimal member, but typically also requiring transverse stiffener plates. Cracks at or near welded joints sometimes occur and represent a potential cause of failure or reduced load rating. We explore here the propagation of guided waves in the plate girder geometry and also the potential for crack detection.

Figure 6. Plate girder geometry

Figure 6 shows a portion of a plate girder, in its customary orientation, and Figure 7 shows the corresponding simulation model with a plane of symmetry at the mid-height of the web. The web and stiffener are 0.64 cm thick,

and the flange is 1.28 cm thick. An S0 wave at a center frequency of 200 kHz was produced by radial forces applied to a circular zone at the mid-height of the web. The simulation domain is symmetric about that surface and all other edges are free.

Figure 7. Simulated z displacement in a plate girder at 20, 30, and 45 µs; pulse center frequency of 200 kHz.

Figure 7 shows the simulated z displacement (in-plane with respect to the web) at t = 20, 30, and 45 µs. When the wave reaches the transverse stiffener, a portion of the wave propagates into the stiffener and another portion continues past the stiffener within the web. (There is also a small reflection from the stiffener, but it is not visible in Figure 7.) When the wave reaches the flange, much of the energy is transmitted into the flange and only a small amount is reflected. These results suggest that reflections will be comparatively small in intact structures. Because free surfaces are perfect reflectors, we can expect significant reflections from cracks, at least when cracks are oriented nearly perpendicular to the direction of wave propagation.

VI. Experimental measurements of wave propagation in a plate girder

61 cm

91 cm

30 cm

15 cm

Figure 8. Measured reflections, at three center frequencies, for model plate girder Reflections occur at joints in intact structures, and it will be necessary to distinguish those reflections from ones that occur at defect locations. We have conducted preliminary experimental studies using a laboratory specimen of a steel plate girder, scaled to realistic cross-sectional proportions. The web is 91 cm deep and 0.32 cm thick, for a height-thickness ratio of 284, and the flange is 10 cm wide and 0.64 cm thick, for a width-thickness ratio of 16; the specimen is 61 cm long, with no stiffeners. A PZT transducer (0.6 cm in diameter) was located on the web 15 cm from a free edge and 30 cm from the upper flange, and reflected signals at three different center frequencies are shown in Figure 8. The transducer was used in pulse-echo mode with a transmit/receive switch. The first reflection after the input circuit recovers from the exciting signal is a strong reflection of the S0 mode from the nearest free surface. This is followed by the A0 mode reflection from that surface, and then a weak reflection from the S0 mode reflected from the flange; a weak reflection from the flange is consistent with the simulations presented above. The remaining reflections that have significant amplitude can be assigned to reflections from the more distant free edge and flange.

61 cm

91 cm

30 cm15 cm

Figure 9. Measured reflections, comparing absence and presence of weld discontinuity

Figure 9 shows results from a second set of experiments, conducted at a center frequency of 379 kHz, in which a PZT wafer-type transducer was located on the web 15 cm from the web-flange joint and 30 cm from the nearest free edge, permitting a relatively clear interpretation of reflections from that joint. A flaw was created by a sawcut, located in the web along the web-flange joint, with a length of 3 cm. Figure 9 shows the measured reflections in the unflawed and flawed cases, and reflections from the flaw region are observed to be significant.

VII. Summary Simulations in two-dimensions and in three-dimensions show relatively weak reflections of Lamb waves from welded joints between steel plates. This is especially true for the web-flange joint in plate girders, where the flange is invariably thicker than the web, a situation that leads to particularly small reflections. Cracks that are oriented perpendicular to the direction of wave propagation, on the other hand, cause strong reflections. These results suggest that guided waves can be useful for flaw detection. The existence of multiple small reflections, particularly in complex geometries, motivates the development of techniques for distinguishing flaw reflections from baseline reflections. Pulse-echo experiments have been performed, using wafer-type transducers mounted on the web, that extend Lamb wave studies from the two-dimensional geometry of flat plates to the three-dimensional geometry of plate girders. The experiments indicate that reflections from the web-flange joints are relatively weak, corroborating observations made in the simulation studies. The experiments also show that a discontinuity in a web-flange weld, oriented normal to the direction of the Lamb waves, produces strong reflections. Acknowledgements We are grateful to the National Science Foundation for support of this work under Grant No. CMS-0329880. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References [1] J.H. Nieuwenhuis, J. Neumann, D.W. Greve and I.J. Oppenheim, “Generation and detection of guided waves

using PZT wafer transducers,” (to be published in IEEE Trans. UFFC). [2] Y. Cho, D.D. Hongerholt, and J.L. Rose, “Lamb wave scattering analysis for reflector characterization,” IEEE

Trans. Ultrasonics, Ferroelectrics, and Frequency Control 44, 46 (1997). [3] M. J. S. Lowe and O. Diligent, “Low-frequency reflection characteristics of the S0 Lamb wave from a

rectangular notch in a plate,” J. Acoust. Soc. Am. 111 (1), Pt. 1, Jan. 2002. [4] V. Giurgiutiu, “Lamb Wave Generation with Piezoelectric Wafer Active Sensors for Structural Health

Monitoring,” Proceedings of the SPIE - The International Society for Optical Engineering, vol. 5056, pp. 111-22 (2003).

[5] For example, see S.W. Kercel, M.B. Klein, B. Pouet, “Wavelet and wavelet-packet analysis of Lamb wave signatures in laser ultrasonics ,” Proceedings of the SPIE - The International Society for Optical Engineering, vol. 4056, pp. 308-17 (2000).

[6] Prada, C., and M. Fink. 1998. “Separation of interfering acoustic scattered signals using the invariants of the time-reversal operator. Application to Lamb waves characterization,” Journal of the Acoustical Society of America, 104(2):801–807.