During the early years of stress-wave theory applied to driven piles, the only way it was done was through the wave equation program. One ran the wave equation, then checked it against field performance. However, a simple blow count check wasn't enough. Were the driving stresses predicted properly? How did the performance of the various components of the driving system (hammer, cushion, etc.) compare with the prediction? Ultimately, the only way these question could be properly answered was through instrumenting the pile (and in some cases, the hammer.)

Although, in his seminal work on stress wave theory in piles, Isaacs had anticipated using the analysis of wave propagation as a substitute for static load tests, the first comprehensive (and successful) attempt at instrumentation was, as we have seen, Glanville et. al. (1938). Subsequent efforts in this direction took place in Sweden, at the Gubbero site in 1960. Both of these efforts photographed the output of an oscilloscope.

The major step in using stress wave theory to analyze piles during driving and to estimate their static capacity was the development of the Case Method. The Case Method is, in part, based on the method of images to solve the wave equation. It can be shown that the wave equation given above can be solved in the form

u(x,t) = f(x-ct) + g(x+ct)

where

• f(x-ct), g(x+ct) = functions of x and t which possess continuous second derivatives

This solution is in the so-called "d'Alembert Form." The solution of the wave equation can be conceptualized as an odd periodic function, the period being defined by the length of the vibrating rod. If this expansion is returned to the physical domain, it shows a series of wave reflections along the rod, as shown above. The wavefront travels from the pile head to the pile toe in a time of L/c, where L is the length of the and c is the acoustic speed of the pile material.

The Case Method compared the pile force and velocity at a given time with a time 2L/c before that. The static and dynamic components were then separated one from another. This method was very simple and could be readily applied in the field, through the measurement of force and acceleration of the pile top using both strain gages and accelerometers. One early paper on these and other devlopments was Soil Resistance Predictions from Pile Dynamics, by Goble, Moses and Rausche, published in 1972.

A more advanced method is the CAPWAP (Case Pile Wave Analysis Program.) Although this technique uses similar instrumentation to the Case Method, the pile is divided up into a series of elements and the reflections from each are analyzed based on their time of return to the pile top. A profile of the shaft resistance distribution is thus obtained. Today this analysis can be done in real time.

Needless to say, other organizations (such as TNO) have developed methods of analyzing the return signals of impact. The result in all cases is once again the use of the hammer, this time in conjunction with stress wave theory and modern measuring techniques, as a measuring tool to estimate the pile's capacity as it is being driven.

Vulcan first encountered dynamic pile analysis offshore, where this technique, like many others, was first applied extensively. A description of the technique from the late 1970's is found here. As with other impact hammers, Vulcan hammers are subject to dynamic analysis in the field. Today pile dynamics are a well-established method of analysing driven piles in the field, and (using special hammers) can also be used with drilled shafts and auger cast piles.

Left: the PDA (Pile Driving Analyser,) which is used to gather and process the data from the pile during driving. With current telemetry, it's no longer necessary for the engineer to be on site to obtain the driving data. But given the variables that take place on any construction site, it's the wise engineer who makes it part of his or her routine to see what's actually going on from time to time.

One further application of stress wave theory in the field is integrity strain testing. This is especially important for drilled shafts, where the actual material integrity of the shaft is not visible from the surface. It can also be used for piling which are suspected to be broken or cracked. There are two variations to this technique:

• Low strain integrity testing, where a small hammer sends down a stress wave and the returning echo is analyzed, much like sonar, and
• High strain integrity testing, which is also used to dynamically measure the pile capacity.