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Why is vibration so hard to predict accurately?

 

Despite years of intensive research and practical applications, predictions in drill and blast scenarios remain as uncertain as ever. Kim Henley and Rob Domotor, of Orica, explain how stress waves work and how vibration is determined in rock formations.

In blasting, stress waves are produced by the detonation of explosives in a blast hole. The rock may be shocked close to the blasthole wall but such shocks rapidly decay and disperse to produce the familiar P- and S-waves (compressional and shear waves), known as body waves. These waves travel throughout the rock mass, compressing and shearing the material. The waves reflect from and refract around the rock structure (joints, foliations, etc) and interact with imperfections, dislocations and the like, causing damage to the rock fabric, resulting in small increases in the local temperature, changes in the grain and crystal structures of the different components and ultimately the growth and coalescence of cracks, leading to fracturing.

The body waves (P- and S-) interact with all surfaces and with all the different joints, bedding planes and the like. When each of these waves hit such surfaces, they undergo mode conversion: an incident P-wave will transmit a P-wave across a joint, for example, but will also produce a reflected P-wave as well as a reflected S-wave, polarised by the interaction with surface. So for every incident P- or S- wave, a combination of P- and S- transmitted and reflected waves are generated, each with their attendant co-efficients (or extents) of transmission and/or reflection and polarisation. There is a similar, even more complex process involving the equivalent refracted components.

These mode conversions determine the nature of the so-called surface waves. A Rayleigh wave (Figure 1) is produced at a rock-air interface (the bench surface or the bench face) and is a combination of the mode-converted waves that manifest on the surface as an elliptically polarised wave that is one wavelength deep but decaying with depth and travelling along the surface for great distances with relatively high amplitude. The Rayleigh waves are responsible for “raising the dust” radiating away from nearly every blast.

Figure 1. Rayleigh waves “raise the dust” radiating away from nearly every blast.

Rayleigh waves are part of a class of waves known as “Lamb waves”. The Rayleigh wave is associated with a single surface whereas the Lamb waves are associated with multiple surfaces in plates or distinct layers within the rock mass. They propagate parallel to the surface throughout the thickness of the material. A Rayleigh wave will transition to a Lamb wave if the plate thickness is less than the wavelength of the Rayleigh wave.

The Rayleigh wave (R-wave) is the most dominant surface wave. These waves can travel for long distances at relatively high amplitudes. While it is true that a structure will be impacted by the R-wave, if the ground between the source and the building has a prominent density gradient, then the P- and S-waves can travel through and across the layers, initially travelling downwards (because the waves are three-dimensional) but then curving upwards by refraction as the density decreases towards the surface. One would then say that damage at the building is caused by the body waves themselves which have effectively moved laterally and “exited” at the surface.

Figure 2. The effect of layered geology, generated by finite element modelling.
Figure 3. The finite element model contains seven horizontal layers with density increasing downwards.

The effect of layered geology is shown in Figures 2 and 3, generated by finite element modelling. In Figure 2 there is one block of rock without layers. The wave leading the rock response at around 400 metres is the P-wave, and the small wave near at 250m is the S-wave. This is followed by the Rayleigh wave, which is the large section at about 150m to 200m. There is a more energy in the R-wave than the P- and S- waves, where white is high amplitude.

In Figure 3, the finite element model contains seven horizontal layers with density increasing downwards. The individual wavefronts cannot be seen but you can clearly see that the high amplitude is spread broadly across the surface, indicating a response that is sustained for a longer period than the case without layers. This complex response is created by the P- and S-waves moving laterally and refracting upward from the lower layers to superimpose to produce a dramatic and complex wave structure that will cause extended vibration at the surface.

NO EFFECT OF VOD ON VIBRATION

Many people ask whether velocity of detonation (VOD) affects vibration. The quick answer is no. Many researchers have attempted to determine whether there is a relationship, but so far none has been found.

Figure 4 shows stress around a detonating blast hole and includes structural discontinuities in the model. You can see the major stress wave generated by the detonation as it travels up the blast hole. It is reasonable to consider that a higher VOD would generate higher stress, and so therefore higher vibration. 

Figure 4. Stress waves travelling up a blast hole.

This may well be true if we were to measure vibration quite close to the blast hole where we would measure vibration without much interaction with the rock mass, as proximity to the explosive charge is quite small. But we don’t measure vibration here. We usually measure it a long distance from the blasthole.

Figure 5 shows the stress of an entire blast and the rock structure around the blast. The stress wave effect on the more distant rock mass shows that the rock response is not driven by a particular stress or strain wave coming from a single blasthole. The stress wave reflects and refracts at every joint as noted above, ensuring that the effect of stress at any point in the rock mass is a complex combination of direct and indirect reflections and refractions from every blast hole. Thus, the distant effects are driven by geological variability, and the combined effect of many holes, so we must consider blast vibration in the context of variable geology, rather than the micro-variations that may occur close to the blast hole.

Figure 5. Stress waves from a blast.

WHY IS VIBRATION HARD TO PREDICT?

Understanding how a structure responds to loading is a classic structural design problem, that every civil engineer deals with, either using direct calculation or by the application of design codes that approximate the problem. The codes and analysis methods assume elastic behaviour of the material, and the connections between structure components are either fixed or have some degree of freedom that may be rotating or sliding. 

A naturally formed indigenous heritage rock formation (Figure 6) in northern Queensland is an isolated column of boulder-sized rocks stacked on top of each other up to 6m high. 

Although the rocks comprising these heritage structures will behave elastically, the connections between them are not defined and it would be presumptuous to think that they could be reasonably defined using classical mechanics due to their random shape and contact surfaces. 

Figure 6. An Indigenous heritage rock formation.

A paper by Shah, Fallah and Louca presents a comprehensive study on counter-intuitive or chaotic dynamic response using prototype discrete parameter models of single or multiple degrees of freedom subject to blast or impact loading.1 The non-linear dynamic behaviour of single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems were studied. 

The paper shows that a SDOF system exhibits a chaotic response below a threshold amplitude but over a particular threshold frequency and amplitude, motion changes to periodic (non-chaotic). In the context of the vibration effects on structures the change to a periodic response represents a resonant or pseudo-resonant response with low damping and therefore high potential for significant amplification. The paper shows that transition of a two-degree of freedom (2DOF) system from chaotic to periodic is independent of frequency but dependent on amplitude.

In this context a chaotic response reflects the complexity of the source vibration and the complex propagation path, as defined by the input vibration signature and is non-resonant. A periodic response is a resonant or pseudo-resonant response that is likely to result in high amplitude structural amplification.

The significance of a transition from chaotic to periodic response as amplitude increases is a possible explanation for the phenomenon that we frequently find where vibration is predictable at low vibration amplitudes but the vibration becomes unpredictably high when it crosses a particular threshold amplitude. The paper shows that the nature of a complex system is such that a small change in the initial amplitude can result in an abrupt change in response and is thus counter-intuitive. You might think that you’re blasting carefully but the odd higher vibration can result in unpredictable results that have no clear explanation.

The following phase portraits from the paper, generated by plotting model displacement on the X-axis and velocity on the Y-axis, show a range of responses. Periodic response from a SDOF, and chaotic/periodic responses from a 2DOF system of first and second order.

The predictable, periodic shape of the SDOF example is clear and represents an approximate constant amplitude waveform over many cycles which is likely to induce amplified response in a structure. The difference between the 2DOF chaotic and periodic examples is subtle but there is increased consistency in the shape of the periodic examples. The main thing to take away from these examples is the greater “randomness” of the chaotic examples, remembering that increased random, or chaotic, behaviour is likely to reduce structural resonant amplification.

Figure 7. A phase portrait of a highly periodic vibration response showing chaotic response at low vibration.
Figure 8. Below about 1.3 mm/s velocity and 0.03mm displacement,
the response of the Radial component is chaotic.

Figure 7 displays the phase portrait of a blast monitor that shows an unusually high response to a blast in the Radial component. This component has an unusual response about three times the amplitude of the other two components and is highly periodic. All components show a highly periodic response, but there is also a lot more chaotic behaviour occurring in the Transverse and Vertical components. 

If our understanding of the paper is correct, then at low amplitude, below the transition from chaotic to periodic response, we should see a more chaotic response (Figure 8). 

The enlarged diagram shows this. Below about 1.3mm per second (mm/s) velocity and 0.03mm displacement, the response of the Radial component is chaotic. In fact, the transition to periodic response is quite sudden, indicated by the banding of the periodic response surrounding the low amplitude chaotic response. This is a case where the Radial component has exceeded the threshold of chaotic behaviour for the structure and its motion has changed to periodic, resulting in significant amplification.

MANAGING VIBRATION

It helps us to recognise that sometimes the vibration we are trying to manage is difficult. Our only response therefore is to be more conservative with our input blast design to account for the apparently random response of a structure, recognising that a transition from chaotic to periodic should be avoided to limit structural amplification. Yet we don’t know the threshold amplitude of the likely transition to periodic response. All we may be able to do is to limit the amount of vibration energy going into the ground to ensure the response of the structure does not exceed the critical transition limit. Of course, it would be very helpful to know at least the magnitude of the critical limit of a geological structure such as this, but the last few million years of its history will remain silent. •

Kim Henley and Rob Domotor are senior specialist technical services engineers at Orica.

REFERENCES & FURTHER READING

1. Shah SKA, Soleiman Fallah A and Louca LA. On the chaotic dynamic response of deterministic nonlinear single and multi-degree-of-freedom systems. Journal of Mechanical Science and Technology 26(6); 2012: 1697~1709. DOI 10.1007/s12206-012-0418-3.

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