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During the early years of stresswave 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 analyse 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(xct) + g(x+ct)
where
 f(xct), g(x+ct) = functions of x and t which possess continuous second derivatives
This solution is in the socalled "d'Alembert Form." The solution of the wave equation can be conceptualised 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 wave front 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 gauges and accelerometers. One early paper on these and other developments 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 analysed 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 analysing 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 wellestablished 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 Analyzer,) 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 analysed, much like sonar, and
 High strain integrity testing, which is also used to dynamically measure the pile capacity.
Note: other papers on pile dynamics and ground vibrations can be found at Mark Svinkin's vulcanhammer.net page. You can also look at a current research project on the subject here.
Other Papers on Pile Dynamics

G.G. Goble, F. Rausche and G. Likins
Ohio Department of Transportation OHIODOT0575
March 1975
A series of research projects have produced a reliable and accurate means of predicting static pile capacity from dynamic measurements. Instrumentation for measuring both force and acceleration at the pile top has been developed and tested. The signal is recorded on analogue magnetic tape using a portable tape recorder. The necessary processing system, both hardware and software, has been assembled so that the recorded data can be analysed completely automatically. The data is first converted to digital form and then a variety of computations are performed and the results plotted. A procedure, using the dynamic measurements, known as the Case Method has been studied which gives capacity predictions in excellent agreement with statically measured values. This method can be applied in the field using a special purpose computer. The concept was fully tested by the project. Methods were also developed for determination of resistance distribution along the pile using measurements made at the top. Extensive correlation between static measurements and dynamic predictions are presented for measurements made in Ohio and also in other states. In all 74 piles were tested. 
Dr. Robert Liang and Luo Yang
University of Akron 
FHWA/OH200708
May 2009
Driven piles are widely used as foundations to support buildings, bridges, and other structures. In 2007, AASHTO has adopted LRFD method for foundation design. The probability based LRFD approach affords the mathematical framework from which significant improvements on the design and quality control of driven piles can be achieved. In this research, reliabilitybased quality control criteria for driven piles are developed based on the framework of acceptancesampling analysis for both static and dynamic test methods with the lognormal distribution characteristics. As a result, an optimum approach is suggested for the number of load tests and the required measured capacities for quality control of driven piles. Furthermore, this research has compiled a large database of pile setup, from which the reliabilitybased approach of FORM is employed to develop separate resistance factors for the measured reference (initial) capacity and predicted setup capacity. This report also provides a Bayesian theory based approach to allow for combining the information from the static pile capacity calculation and dynamic pile testing data to improve pile design process. Specifically, the results from dynamic pile tests can be utilized to reduce the uncertainties associated with static analysis methods of pile capacity by updating the corresponding resistance factors. This research has also developed onedimensional wave equation based algorithm to interpret the High Strain Testing (HST) data for the estimation of the shaft and toe resistance of driven piles. The closed form solution is obtained for determining the Smith damping factor and the static soil resistance. Finally, a set of new wireless dynamic testing equipment (both hardware and software) is developed for more efficient dynamic pile testing. 

James Long and Andrew Anderson
Illinois Center for Transportation, University of Illinois at UrbanaChampaign
FHWAICT14019
September 2014
A dynamic load test program consisting of 38 sites and 111 piles with restrikes was conducted throughout Illinois to improve the Illinois Department of Transportation design of driven piling. Pile types included steel Hpiles and closedended pipe (shell) piles. Piles were driven into all soil types including clay, silt, sand, shale, and limestone. Predictive methods for estimating pile capacity were investigated and include the KIDOT (static) method, WSDOT (dynamic formula), WEAP, PDA, and CAPWAP. Pile capacities were taken as the capacity estimated using CAPWAP for beginning of restrike conditions. Piles were monitored during initial driving. Piles were redriven several days later to assess the amount of setup to assess the effect of time, pile type and soil type. Restrikes were conducted typically between 3 15 days after initial driving. Modifying WSDOT to include effects of setup explicitly with specific equations (Skov and Denver, 1988) for time dependent setup was not any more precise than the original WSDOT formula with adjustments for pile type. Accordingly recommendations are made for adjusting WSDOT estimates based on whether the pile is an Hpile or a shell pile. Adjustments were made to the simplified stress formula (SSF) to refine predictions of stresses in driven H and Shell piles driven with diesel hammers. Resistance factors were determined using the First Order Second Moment method for the static method (KIDOT) and the dynamic formula (WSDOT). Pile types included Hpiles and shell piles for both end of driving conditions and for beginning of restrike. Resistance factors were also determined for WEAP and PDA. These resistance factors were determined using the CAPWAP (BOR) capacity as the static capacity for the pile, although it is preferable that the resistance factors be based on static load test. Accordingly, adjustments were made to the resistance factors accounting for the average agreement between capacity determined by CAPWAP(BOR) and capacity determined with a static load test 

Kevin F. Kett and G. Thomas McDaniel
ASCE Florida Section Annual Meeting
October 1987
Case histories of seven solid, square, prestressed, precast concrete piles driven into fine sand In Florida are presented. These piles were evaluated using two static prediction methods, (1) the Florida DOT Pile Capacity Method (FDOT Bulletin 121A, and (2) The Federal Highway Administration Nordlund Method; and a dynamic prediction procedure (1) the Pile Driving Analyzer with the CAPWAPC wave equation computer model. Both axial tension and compression capacities were evaluated by the presented methods and compared to static pile load tests carried to failure. The pile ranged in size from 12 to 20 inches square. These piles were driven into very loose to very dense fine sands, clayey fine sands and silty fine sands. The prediction methods which correlated favourably with the static load test results are presented and discussed. 
(This document has a blog page, where you can leave your comments.) 
Don C. Warrington
May 2013
This article is an overview of the current state of both forward and inverse analysis of wave propagation in piling. It begins with a summary of the typical acceptance procedure for the wave equation as applied to (primarily) driven piles. It then defines and describes what are forward and inverse methods, outlining criteria which are important for success. After this the governing equations are discussed, both undamped and damped (Telegrapher's) wave equations, and why it is important to consider the latter as the true governing equation for pile dynamics. This is followed by a discussion of explicit and implicit methods and how they are (and might be) applied to the problem at hand. The difference between finite difference and finite element methods is discussed, and how each has been applied in either a onedimensional or twodimensional way. Finally the issue of rheology is examined. The central problem with dynamic analysisthe inability to separate static and dynamic resistance by the basic inverse methods availableis discussed in detail. 

G.G. Goble and F. Rausche
Institution of Civil Engineers, Numerical Methods in Offshore Piling, 1980
The CAPWAP analysis is performed on data obtained during the installation of a conductor pipe. Dynamic soil are derived and are used for analysing the drivability of the jacket piles. A case study is described in which the driving statistics of jacket piles were predicted and compared with the results obtained during platform installation. 

Frank Rausche
PhD Dissertation, Case Western University
1970
An automated prediction scheme is presented which uses both measured top force and acceleration as an input and computes the soil resistance forces acting on the pile during driving. The distribution of these resistance forces acting along the pile is also determined. Shear and dynamic resistance forces are distinguished such that a prediction of total static bearing capacity is possible. Using the shear force prediction a static load versus penetration curve is computed for comparison with the result from a corresponding field static load test.
The method of analysis uses the traveling wave solution of the onedimensional, linear wave equation. As a means of calculating the dynamic response a lumped mass pile model is used and solved by the Newmark betamethod.
Using stress wave theory two simplifiedmethods are developed for predicting static bearing capacity from acceleration and force measurements. These methods can be used during field operations for construction control when incorporated in a special purpose computer. The automated prediction scheme and simplified methods are applied t o 24 different sets of data from full scale piles. The piles were all of 12 inches diameter steel pipe with lengths ranging from 33 to 83 feet. Also, 24 sets of data from reduced scale piles are analysed by the simplified methods. All predictions are compared with results from static load tests. Correlation is very good for piles driven into noncohesive soils. For cohesive soils better agreement with static load measurements are obtained than from existing methods. As a check on the assumed soil response to both pile displacement and velocity results from measurements taken at the pile tip are investigated and discussed. Further, an approach to pile and hammer design is described using the results of stress wave theory. 

