It's easier to understand where we're going if we understand where we've been. This paper discusses just the wave equation as applied to piles.

The Wave Equation In General

The classical one dimensional wave equation is given by the formula

u(x,t)_tt = c² u(x,t)_xx

where

• u(x,t) = displacement as a function of distance and time
• c = speed of sound in propagating medium

This equation is a second order, hyperbolic, partial differential equation. It can be solved by a number of methods, both closed form and numerical. Some of these as they apply to driven piles are discussed below.

Method of Images (d'Alembert's Solution)

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

Wave Reflection Chart (after Uto et. al, 1985)

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 method of images is based on this concept, because it seeks to solve the wave equation by considering the effects of the periodic transmissions and reflections of the stress wave generated by the hammer along the pile. In doing this it attempts to avoid the complexities of other closed form solutions. The method of images has been influential throughout the development of stress wave theory as it is applied to piles, from the earliest efforts to the pile monitoring techniques.

Dynamic Formulae

Piles have been driven in the ground to support structures in weak soils since the beginning of civilization. For millennia it was a process done by experience and almost always with wood piles. The first attempt to model the dynamics of pile driving and to make this modeling useful to practitioners resulted in the dynamic formulae. These formulae use Newtonian impact mechanics to model the motion of the pile; this results could generally be expressed in a simple formula, which could be readily applied to the work at hand.

The most widely disseminated dynamic formula is the Engineering News formula, although many others have made their way into codes and other standards of practice (Hiley, Gates, Danish, etc.) There are many others and variations even of these as well. Many foundations have been installed using these formulae as a guide.

Although the dynamic formulae have been pilloried extensively since the wave equation became practical to use, their two main weaknesses are

• modeling of the pile as one rigid mass
• inadequate modeling of the soil as it interacts with the pile.

These weaknesses and others were not as apparent when wood piles installed using drop hammers was the norm in pile driving. With the introduction of concrete and steel piles these deficiencies became critical, especially when concrete piles began to exhibit tension cracking, a phenomenon the dynamic formulae could neither anticipate nor quantify.

Isaacs' Work

The use of the wave equation (or stress-wave theory) to model impact pile driving began in Australia with the work of David Victor Issacs (1931). The dynamic formulae had been developed primarily with timber piles in mind; with the growing usage of concrete piles, it became apparent that, because of the length and properties of timber pile, the dynamic formulae (with their assumption that the pile is a rigid mass) would not be sufficient for concrete piles. Isaacs started out by reviewing the dynamic formulae. Part of his review included a discussion of the factor of safety, where he makes a statement that is still relevant:

It should be remembered, however, that these are not true factors of safety, but include a "factor of ignorance." The author suggests that when the ultimate resistance of any pile has been determined, in fixing the factor of safety...the most unfavorable conditions possible in the supporting strata should be judged (the range of conditions possible being narrowed with better knowledge of the subsurface conditions and of the possibility of disturbance from extraneous sources) and a proportion of the factor of safety -- a "factor of ignorance" -- then allowed in respect to these possible conditions, the manner of determining the ultimate load, and the type of loading to be borne. The remaining proportion of the factor of safety -- or true margin of safety -- should be approximately constant for all classes of loading and foundation conditions involving the same value of loss in case of failure; and the overall factor of safety...will then be equal to the product of the true factor of safety with the "factor of ignorance." (p. 305)

After this, he describes an experiment where rods are impacted against each other in a pendulum setup. As the rods were lengthened, the behavior of the rods deviated more and more from Newtonian impact theory.

He then went to develop an integration technique that is best described as a semi-graphical one. He developed a mathematical model based on the successive transmission and reflection of waves (similar in principle to the method of images.) A sample solution is given below, in this case showing multiple impacts.

Graphical Solution of the Wave Equation (after Isaacs, 1931)

He then constructed a drafting machine to draw out the solution, a diagram of which is shown below.

Graphical Machine for Wave Equation Solution (after Isaacs, 1931)

He was then able to solve for the stresses and displacements of the pile during driving. Isaacs developed a set of formulae and charts to make his results accessible for the analysis of piles.

In the course of the investigation, Isaacs dealt with a number of questions that would become central to stress wave analysis of piles, including tension stresses in concrete piles, the effect of ram weight (he concluded that to a point a heavier ram reduced tension stresses,) and the effect of cushion material stiffness and drive cap weight.

Issacs' work also revealed the computational complexity of stress wave analysis, a complexity that insured the dominance of dynamic formulae in pile analysis (with all of their serious limitations) for another half century.

The Work of Glanville et. al. (1938)

After Issacs' work, the center of work on the subject shifted to the United Kingdom, with the extensive study commissioned by the British Building Research Board under the direction of (later Sir) W.H. Glanville and his colleagues. This study was one of the first comprehensive studies on stress waves in piles in general. It was occasioned by the problems encountered in the breakage of concrete piles during driving, both at the top and the toe. The wave equation and d'Alembert's solution were used to develop equations to estimate the stress in the pile during driving, using the method of images. Because of the complexity of the equations, the results were reduced to a series of charts where a quantity of dimensionless stress was plotted against the ratio of hammer weight to pile weight. The charts could then be used to estimate pile stresses and resistance. The charts were applicable to concrete piles only, and this was and is a serious limitation to such solutions, because they were applicable to a limited universe of piles.

In addition to developing a solution to the wave equation, the authors continued Isaacs' (1931) work in addressing technical issues and experimental techniques that have enduring interest in pile dynamics. These include the instrumentation and data collection on stresses and forces in piles, including remote data gathering through "portable" equipment in a trailer, further research on the effect of the hammer cushion on the generation and effect of the pile stress wave (these were included in the analytical work,) drop tower testing on cushion material to determine the cushion stiffness, and further work on the relationship of ram weight to pile weight and cross section. An example of the data collected is shown below, which shows recorded tension stresses in the midpoint of the pile during driving.

Recorded tension in the pile midpoint (in psi, after Glanville et. al., 1938)

After this study the Second World War interrupted further research; moreover, the difficulties that this study encountered in developing a readily usable prediction technique ended the possibility of using a closed form solution for this application. Research has continued to the present on closed form solutions; the most recent effort in this regard is that of Warrington (1997), which combines a semi-infinite pile solution (see below) and Fourier series to overcome some of the difficulties encountered in the past.

Semi-Infinite Pile Solutions

Before we get into the numerical methods, one special type of solution of the wave equation is the solution of the wave equation for semi-infinite piles. In these solutions, the pile is modeled as a semi-infinite bar, struck at one end. The result of this analysis is to model the interaction of the hammer and the pile only, generally without direct consideration of soil interaction or of numerical problems associated with hammer-pile modeling.

The central concept in semi-infinite pile modeling is mechanical impedance, which is defined as

where

• Z = mechanical impedance of the pile, N-sec/m
• F = force for a given pile cross section, N
• V = particle speed in the pile, m/sec

For a bar of uniform cross section, the mechanical impedance is given by the equation

where

• A = cross-sectional area of the pile, m²

Because there are neither other reflections nor the movement of the pile to consider, when this response is mapped to the distance domain along the pile, it represents the entire response of the system. Strictly speaking this model is only applicable to piles with no dampening or soil elasticity along the shaft, a point which is frequently overlooked.

The most comprehensive solution of the hammer force-time curves using this theory was carried out by Deeks (1992). His main objective was to use these results to evaluate numerical methods of analysis for piles, an important application for closed form methods. Deeks also considered losses in the cushion material as viscous losses, which give the possibility of analyzing variations in the loading rate of the cushion material, as opposed to the static one presently used with finite difference wave equation analyses.

Enter the Numerical Methods

With the difficulties apparent in applying the closed form solutions, the obvious alternative was to use a numerical method to model stress waves in piles and thus the pile behavior itself. Today we say "obvious" but in the 1940's and 1950's this was not the case, primarily because the practical application of numerical methods required the use of a digital computer, which was just getting its start as well. Moreover geotechnical engineering in general was slower than other specialties to adopt advanced analytical techniques, primarily because the ground itself presented complexities that are in reality only now beginning to be modeled on a reasonable basis. Experience and judgment, essential elements in any engineering practice, were and are especially important in geotechnical applications.

It is with this background that we need to see how remarkable the accomplishment of Smith (1960) really is. E.A.L. Smith was the Chief Mechanical Engineer for Raymond International, at the time the world greatest organization for the installation of piles; by the time his historic paper had been published, he had already retired. The basic elements of Smith's technique are as follows:

• Division of the pile into a series of springs and masses. The hammer was modeled as a mass with a hammer cushion spring (or capblock, to use the Raymond terminology.) The driving accessory (or follower) was also modeled as a mass, with provision for another cushion for the pile if needed.
• Integration of the model using a first order finite difference technique.
• Modeling of the hammer and pile cushions using a "static" hysteresis technique, defined by an elastic spring constant during compression and modified by a "coefficient of restitution" during rebound.
• Modeling of the soil as a combination of displacement dependent springs and velocity dependent dampers. These were applied both along the pile shaft as the actual soil resistance distribution required.
• Modeling of the non-linearities of the soil. The soil was given a "yield limit" (a quake); after this time the non-dynamic resistance was constant. The model could simulate yielding in some areas of the pile without requiring it in others.

Smith made many of the assumptions for his model (especially for the soils) with a minimum of theoretical backing; nevertheless, the soil model he proposed is still standard in many wave equation programs today. This is a tribute to the durability and basic soundness of his proposal.

Even with the advanced state of Smith's model, it was several years before this model was disseminated to practitioners. The first version of Smith's model that was made available to a wider audience was the TTI (Texas Transportation Institute) program, developed at Texas A&M University by L.L. Lowery and T.J. Hirsch (Lowery et. al, 1969; Hirsch et. al., 1976.) In addition to making Smith's model available, many of the assumptions behind the Smith model were examined, some experimentally, to improve the quality of the input and thus confidence in the output. The TTI program is still available and is used especially with offshore oil and gas platform piling.

When Smith developed his wave equation method, he only included hammers with relatively short rams and cushions. This is principally because Raymond primarily used air-steam hammers at the time. Neither the Smith nor the TTI programs had anything past a rudimentary modeling of the combustion complexities of diesel hammers, which became popular in the U.S. in the 1960's. This deficiency was corrected with the WEAP program, developed by George Goble and Frank Rausche (Goble and Rausche, 1976). In addition to advancing the modeling of the diesel hammers, WEAP and its successors gave the wave equation for piling the one thing it needed more than anything else: an organization (GRL) prepared to actualize it through support and dissemination. An example of this support for the wave equation itself was with WEAP86, which included a hammer database and PC capability (Goble and Rausche, 1986). This support became more critical with field monitoring techniques.

There are other wave equation codes in use today, such as TNOWAVE, put out by the TNO organization in the Netherlands. In addition to the finite difference technique, there have been finite element methods used, either one dimensionally or with the surrounding soil modeled.

Field Monitoring Techniques

One of the signal characteristics of geotechnical engineering is the high degree of uncertainty created by the use of soils as an engineering material. The wide variability of soils (especially with deep foundations) and the difficulty inherent in modeling their behavior makes some kind of in situ verification essential.

With driven piles, the dynamic formulae gave the industry just such verification. By relating the blow count of the hammer per foot (or inch, meter, etc.) of pile penetration to the resistance of the pile, the capacity of piles could be verified. This put the pile hammer in an unusual position as both an installation tool and as a measuring device.

Although the wave equation enabled prediction of performance, relating that prediction to the actual capacity or resistance of the pile involves more than just comparing the wave equation to the static load test. From the beginning some kind of instrumentation was felt necessary to learn more about the pile's behavior during driving and to make some intelligent surmises about what that behavior might be after driving and during use.

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 (Fellenius, 1996). 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 (Goble et. al., 1980). This 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.

A more advanced method is the CAPWAP (Case Pile Wave Analysis Program) (Rausche et. al., 1985.) 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.

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.

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.

Conclusion

The use of the wave equation for piling has come a long way from the early years of Isaacs and Glanville, to the point where we can estimate the capacity of a pile using these techniques. However, there is still much work to be done in this field, to improve our confidence in the results and to have as our result more economical and reliable foundations.

References and Further Reading

DEEKS, A.J. (1992) Numerical Analysis of Pile Driving Dynamics. Ph.D. Thesis, University of Western Australia.

FELLENIUS, B.H. (1996) "Reflections on Pile Dynamics." Proceedings of the Fifth International Conference on the Application of Stress-Wave Theory to Piles, 11-13 September 1996, Orlando, Florida.

GLANVILLE, W.H., GRIME, G., FOX, E.N, and DAVIES, W.W (1938). "An Investigation of the Stresses in Reinforced Concrete Piles During Driving." Department Sci. Ind. Research, British Building Research Board Technical Paper No. 20.

GOBLE, G.G., and RAUSCHE, F. (1976) Wave Equation Analysis of Pile Driving, WEAP Program. U.S. Department of Transportation, Federal Highway Administration, Washington, DC. Report FHWA-IP-76-13 (4 Vols.)

GOBLE, G.G., RAUSCHE, F., and LIKNS, G.E., Jr. (1980) "The Analysis of Pile Driving -- A State of the Art" Proceedings of the International Seminar of the Application of Stress-Wave Theory on Piles, Stockholm, Sweden, 4-5 June.

GOBLE, G.G., and RAUSCHE, F. (1986) Wave Equation Analysis of Pile Driving, WEAP86 Program. U.S. Department of Transportation, Federal Highway Administration, Washington, DC. Report DTFH61-84-C-00100 (4 Vols.)

HIRSCH, T.J., LOWERY, L.L., and CARR, L. (1976). Pile Driving Analysis -- Wave Equation User's Manual, TTI Program. U.S. Department of Transportation, Federal Highway Administration, Washington, DC. Report FHWA-IP-76-14 (4 Vols.)

ISAACS, D.V. (1931) "Reinforced Concrete Pile Formulae." Journal of the Institution of Engineers Australia, Vol. 3, No. 9, September, pp. 305-323.

LOWERY, L.L, HIRSCH, T.J., EDWARDS, T.C., COYLE, H.M. and SAMSON, C.H. (1969). Pile Driving Analysis -- State of the Art. Research Report 33-13. College Station: Texas Transportation Insititute.

RAUSCHE, F., GOBLE, G.G., and LIKINS, G.E., Jr. (1985) "Dynamic Determination of Pile Capacity." Journal of Geotechnical Engineering, Vol. 111, No. 3, March. New York: American Society of Civil Engineers.

SMITH, E.A.L. (1960) "Pile-Driving Analysis by the Wave Equation." Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers. Vol. 86, No. EM 4, August.

TOLSTOV, G.P. (1962) Fourier Series. Englewood Cliffs, NJ: Prentice-Hall, Inc..

UTO, K., FUYUKI, M. and SAKURAI, M. (1985) "An equation for the Dynamic Bearing Capacity of a Pile Based on Wave Theory." Proceedings of the International Symposium on Penetrability and Drivability of Piles, San Francisco, 10 August 1985. Tokyo: Japanese Society of Soil Mechanics and Foundation Engineering.

WANG, Y.X. (1988) "Determination of Capacity of Shaft Bearing Piles Using the Wave Equation." Proceedings of the Third International Conference on the Application of Stress-Wave Theory to Piles, pp. 337-342. Vancouver: Bi-Tech Publishers.

WARRNGTON, D.C. (1997) Closed Form Solution of the Wave Equation for Piles. Master's Thesis, University of Tennessee at Chattanooga.