Introduction
Wave propagation in impact driven piles has long been recognised, and is now used routinely to predict, modify and verify driving stresses and ultimate capacity of driven piles. Unfortunately, the numerical methods that were adopted early in the actualisation of the theory have turned the whole subject into something of a “black box” affair for most civil engineers. This monograph seeks to dispel some of the mystery behind them and give an understanding of how waves are propagated in piles and in turn how this can give us important information as to the performance of the foundation, both during installation and during service life.
The Wave Equation
The basic assumption of all wave mechanics in piles is that the pile responds to impact according to the onedimensional wave equation. Let us begin by assuming that equation to be undamped
Equation 1:
where
c = Acoustic Speed of Pile Material, m/sec =
u(x,t) = Displacement of Pile Particle, m
t = Time from Zero Point, seconds
x = Distance from Pile Top, m
E = elastic modulus of the material, Pa
ρ = density of the material, kg/m^{3}
The initial conditions are
Equation 2:
and
Equation 3:
where
f(x) = Initial Displacement Distribution in Pile, m
g(x) = Initial Velocity Distribution in Pile, m/sec
The initial conditions reveal the first difficulty in applying the wave equation as we would, say, in acoustics, to a string. The pile is assumed to be at rest at the initial time. With null initial conditions, a standard Fourier series solution is impossible. The excitation for the pile comes from one end of the pile in the form of the hammer impact, but its effects do not come into play until t > 0 and thus are not an initial condition.
With these and other difficulties, what is the best way to begin our understanding of wave propagation in piling? The best way is to start with the simplest case: the semiinfinite pile.
SemiInfinite Pile Theory
As the name implies, the semiinfinite pile is a pile with ends at x = 0 and x = ∞. (Some contractors will tell you that some piles they have driven “took forever,” but these are not semiinfinite piles.) The advantage of semiinfinite piles is that, in the undamped state, they have no reflections back to the pile head, which simplifies the analysis.
The first result of semiinfinite pile theory is that, assuming the displacementtime curve at the pile top induced by the ram impact is defined by f(t), the response of the pile is
Equation 4:
where = Heaviside Step Function
This is half of the familiar d’Alembert solution of the wave equation; the other half disappears because the pile is semiinfinite.
That, in turn, leads to the next question: how does the displacementtime curve relate to the forcetime curve? For semiinfinite piles, it can be shown that
Equation 5:
This means that the strain and the particle velocity are directly proportional.
Since from elasticity the force in the pile at any point can be computed by the equation
Equation 6:
Where A = crosssectional area of the pile, m^{2}
Equation 5 and Equation 6 can be combined to yield
Equation 7:
If we define the pile impedance as
Equation 8:
Where Z = pile impedance, Nsec/m
Equation 7 can be rewritten as
Equation 9:
The pile impedance is an important quantity for the following reasons:
 For purposes of modelling the hammer system (ramcushionanvilcap,) the entire pile can be modelled as a velocitybased dashpot, since u_{t} is the particle velocity in the pile. This allows the generation of forcetime curves for various hammer configurations and initial conditions. The model can be closed form or numerical.
 The impedance is a measure of the way in which piles transmit force. Thus, a high impedance pile can transmit force with relatively low strains, stresses and velocities. A low impedance pile requires higher strains, stresses and velocities for the same force. This can affect the drivability of a pile, both in terms of the driving stresses and the pile penetration.
Modelling the Pile Hammer

Now that semiinfinite pile theory makes modelling the pile relatively simple, we can turn to modelling the hammer. The first issue we must address is the configuration of the hammer itself. Several possible configurations are shown in Figure 1, although this is certainly not exhaustive. All of these can be solved as single or twodegree of freedom vibration problems, either in closed form or numerically.
The solution for case (a) is simple, and can be given as
Equation 10:
Where
V_{o} = velocity of the ram at impact, m/sec
M = mass of the ram, kg
Unfortunately it is the rare pile which is struck by the ram directly, and generally the distributed mass and elasticity of the ram come into play. Cushionless hammers such as these are certainly used to drive piling but are beyond the scope of this article.
More realistic hammers are shown in (b), (c) and especially (d). Before we come to analyse these, a few definitions are in order.
The first is the hammer impedance:
Equation 11:
Where
Z_{s} = Hammer impedance, Nsec/m
K = hammer cushion stiffness, N/m
From this we can define the pilehammer impedance ratio as
Equation 12:
The second is the theoretical peak force of the ramcushion system. If the pile top or cushion is rigid, the forcetime curve will be a halfsine wave with the peak force of
Equation 13:
The third is the natural frequency of the ramcushion system:
Equation 14:
And its inverse, the ramcushion period
Equation 15:
Hammer Mass and Energy
All of these quantities introduce an important concept that most engineers overlook in evaluating a pile hammer: the relationship between the mass and the way in which the hammer outputs energy.
The energy output of a hammer is given by the equation
Equation 16:
A given energy can thus be achieved by one or two methods: increase the ram and decrease the striking velocity, or the opposite. Since it increases by the square of the velocity, it seems simpler to increase the velocity, achieved by increasing the ram stroke, the downward assist or both.
However, the object of impact pile driving is for the hammer to generate a peak force high enough to move the pile. Equation 13 indicates the same thing as Equation 16, i.e., the easier way to move the pile is to increase the impact velocity, assuming the hammer cushion stiffness is the same. For many piles (esp. steel piles) this is a desirable result.
The one situation where this rule requires modification is with concrete piling. Here, controlling stresses (both compression and tension, as we will see) is more critical due to the nature of the material. Looking at Equation 15, it is frequently advantageous to increase the mass of the ram and produce a longer, more sustained blow which will deliver the energy to the pile and at the same time produce stresses within the limit of the material. (It is also possible to use lower cushion stiffness, but a ram mass increase is easier to implement on a verifiable basis.)
Modelling the Pile Hammer
Turning to the other hammer models, Equation 10 suggests that a closed form solution for the configurations shown in Figure 1 would be simple. Such is not the case. Let us look at the configuration in Figure 1(c) to see how this works out.
The equations of motion are, for the ram and cap respectively,
Equation 17:
and
Equation 18:
where
x_{r} = Displacement of ram, m
x_{t} = Pile Top Displacement, m
m = mass of pile cap, kg
The solution for this equation is
Equation 19:
Where
α_{1}, α_{2}, α_{3} are dimensionless coefficients based on system parameters
m’ = M/m = mass ratio of ram and cap
Obviously an analytical solution to this problem is very involved, and Equation 19 as it stands does not take into consideration the fact that the ramcushion, the capcushion and the cappile interfaces are all inextensible.
It is interesting to note that the maximum pile top force for Figure 1(c) (Warrington (1987)) can be estimated using the following equation:
Equation 20:
Where F_{3} = maximum pile top force for the case described by Figure 1(c), N.
Solution for Finite Piles
Now we can take the next step: consider a pile of finite length. The system we will consider is shown in Figure 2. The hammercap system is the same; we could have used any of the alternatives shown in Figure 1 or another system. The pile has a finite length L and a toe spring at the bottom with stiffness k_{p}. In addition to the assumptions stated in Figure 2, the pile is assumed undamped and without external spring constant along the shaft, thus Equation 1 applies.
The first question that needs to be considered is this: how is it possible that theory developed for an infinitely long pile be applicable to one of finite length? To answer that question, since we defined a natural frequency for the hammer system, let us also define one for the pile system as follows:
Equation 21:
where
ω_{2} = Natural frequency of pile, rad/sec
L = Length of Pile, m
Conversely we can define a pile period as follows
Equation 22:
Where t_{p} = pile period, sec.
If we examine Equation 4 carefully, we realise that the entire wavefront travels down the pile at the acoustic speed c of the pile material, which reaches the toe at the time of the pile period. In most cases, the waves are reflected from the toe and take another ram period for the wavefront to return to the pile head. Thus, the pile head is unaffected by the wave activity in the pile for the time 2t_{p}, or “2L/c,” as it is commonly referred to. In this way semiinfinite pile theory is meaningful in real piles. Obviously longer piles will behave more faithfully to the semiinfinite model than shorter ones do; wave propagation in piles is more significant in longer piles.
The second question is this: how do we deal with the flexible pile toe? Most closed form solutions of the wave equation that engineers are familiar with assume that the ends of the string or bar are fixed or free. For closed form solutions, boundary conditions such as this dictate the use of transcendental functions to solve for the eigenvalues, but ultimately this situation points to the need of numerical methods in practical application.
Example of Wave Propagation in Undamped Pile
Consider the case of a steel pipe pile 50 m long, 1000 mm in diameter with a wall thickness of 40 mm. The hammer has a ram with a mass of 15,000 kg, a stroke of 1500 mm and a mechanical efficiency of 80%, which results in an impact velocity of 4.85 m/sec. The cap has a mass of 3000 kg and the cushion has a stiffness of 2.45 GN/m.
Let’s begin by computing a few parameters.
First, the crosssectional area of the pile:
The acoustic speed of the steel (Equation 1):
The pile impedance (Equation 8):
And finally the pile period (Equation 22):
Or
For simplicity’s sake, we will define the stiffness of the pile toe (soil stiffness) in terms of the static stiffness of the pile. The static stiffness of the pile (familiar to most geotechnical engineers through its use in Davisson’s Method) can be computed as follows:
Equation 23:
where
K_{p} = Pile static stiffness, N/m
K_{s} = Soil Toe Spring or Elastic Constant, N/m
We will use a closed form solution for the wave equation (Warrington (1997).) This closed form solution uses Equation 4 for t < t_{p} and a Fourier series after that. The eigenvalues for the Fourier series are
Equation 24:
The simplest way to present the results is through a series of plots.
The first set of plots will be for the assumption that the pile toe is a fixed end. In this case K_{s} = ∞ and K_{s}/K_{p} = ∞. The first plot will be for the displacements.
To understand this plot, from front to back is the xaxis of the pile (geotechnical engineers would say the “zaxis,” to be proper.) The front of the graph represents the pile head and the back of the graph represents the pile toe. From left to right is progressing time. At the left is zero time, at the far right is the maximum time of 4L/c. Thus we are graphing two cycles of wave propagation in each pile.
At the pile head we see a “set” of the pile head which is reversed around 2L/c to a value which is opposite of the original “set.” This “set” is the dynamic elastic compression of the pile. When a hammer strikes a pile, it elastically compresses the pile at the head, and the energy in that compression is then transmitted down (and reflected back up) the pile. This elastic compression is different from the static elastic compression we are familiar with in Davisson’s Method and pile settlement methods; this confusion is one of the key weaknesses of dynamic formulae.
We also note the flat portion of the graph at the left end. This represents the part of the pile which has not been reached by the wavefront immediately after impact. At the pile toe, for example, the displacement remains at zero until time L/c, as wave theory predicts.
Let us once and only once turn the graph around and see the displacements from the toe end upward.
We can see that the boundary condition of a fixed pile toe has been satisfied.
Now we turn to the pile stresses.
Note first the somewhat sinusoidal profile at the pile head which is projected down/moves down the pile. This is the result of the forcetime curve of the hammer. I say “somewhat sinusoidal” because, if the cap and pile head were immovable, it would be sinusoidal. The curve is modified by both the effect of the cap movement and the impedance of the pile, which acts as a “dashpot.”
After the wave reaches the toe, it is reflected back up to the pile head during time L/c < t < 2L/c. It’s interesting to see, however, that during the reflection process the pile stresses and forces are magnified by the superposition of the incident and reflecting wave. This “wave doubling” is an important phenomenon. Fixed end cases such as this are meant to simulate the condition of end of driving, high resistance, low blow count pile driving, and this doubling is an assist in keeping the pile moving during driving.
When the wave comes back to the top at 2L/c, it’s reversed again. Notice, however, that the wave changes from a compression wave to a tension wave. This is because the closed form solution we use assumes that the pile head is free after L/c. This assumption is not entirely realistic because of the hammer, but it illustrates the nature of reflections in wave propagation:
 Reflections from a fixed end have the same sign as the incident wave, i.e., compression waves are reflected as compression waves, while tension waves are reflected as tension waves.
 Reflections from a free end have the opposite sign as the incident wave, i.e., compression waves are reflected as tension waves, while tension waves are reflected as compression waves.
 Reflections from cases between the two have an intermediate nature.
This is important, as we will see.
Let us proceed to the intermediate case. We will use the pile toe stiffness ratio of K_{s}/K_{p} = 0.5. First the pile displacements:
We see a stepwise progression at both the pile top and pile toe. In real piling, assuming the toe (and shaft) resistance are mobilised, this will translate into permanent set for the pile, but we have a purely linear model here.
We see a basically “free end” type of tension reflection in the pile, but the stress intensity is not completely reversed. We also see a compression reflection after 2L/c that is actually greater than the original wave. This complex interaction of boundary conditions and wave propagation is what we see in actual pile driving.
Now we should turn to the last case, the free pile toe, where K_{s} = 0 and K_{s}/K_{p} = 0.
Since there is no resistance at the toe, the pile is actually moving downward as a mass, with the wave propagation modulating the movement.
What we have here is basically both ends free (except for the initial impact at the head.) Thus every reflection reverses the sign of the wave: all downward waves are compressive, all upward waves are tensile.
What’s also important to note here is that the magnitude of the reflected wave is equal to the incident one. Thus the tension stresses the pile experiences are the same magnitude as the compressive ones. This is easier to see if we look at the plot from a straighton end view (Figure 10.)
The free pile toe condition roughly simulates the pile in the early stages of driving, when soil resistance (both shaft and toe) is at a minimum. Tension stresses in piling are especially undesirable in concrete piles, especially reinforced concrete piles (ones with no prestress.) It was, in fact, tension cracking in the middle of precast concrete piles which first led to serious consideration of wave mechanics in piles and Isaacs’ pioneering work.
Today these are controlled by reducing the stroke of the hammerand thus the impact velocityin the early stages of driving. This also reduces pile run during low resistance driving, which can be very destructive on the pile driving equipment as well.
Wave Equation in Practice
Note: most of the following figures and the example in the Case Method section are taken from this reference. The methodologies are those of GRL and PDI.
Although the foregoing considerations help us to understand wave propagation in piles, they cannot be applied to practical pile driving problems because they do not take into consideration important physical properties of these systems. Some of these include but are not limited to the following:
 Existence of dampening, both at the toe, along the shaft, and in all of the physical components of the system. In theory, inclusion of distributed spring constant and dampening along the shaft could be simulated using the Telegrapher’s wave equation, but other factors make this impractical also.
 Nonlinear forcedisplacements along the toe and shaft, and in the cushion material. Exceeding the “elastic limit” of the soil is in fact one of the central objects of pile driving.
 Nonuniformity of soils along the pile shaft, both in type of soil and in the intensity of the resistance.
 Inextensibililty of many of the interfaces of the system, including all interfaces of the hammercushionpile system and the pile toe itself.
 Nonuniformity of the pile cross section along the length of the pile, and in some cases the pile changes materials.
 Slack conditions in the pile. These are created by splices in the pile and also pile defects.
 With diesel hammers, the forcetime characteristics during combustion are difficult to simulate in closed form. (It actually took around fifteen years, until the first version of WEAP was released, to do a proper job numerically.)
 Unusual driving conditions, such as driving from the bottom of the pile or use of a long follower between the hammer and the pile head.
All of these conditions require the use of numerical methods for proper solution.
Figure 11 illustrates the application of numerical methods to the solution of the wave equation for piles. The pile is divided into segments and the stiffness and dampening for hammer, cushion, pile and soil is included. This allows for a much more realistic simulation of a pile during driving than is otherwise possible.
The method was first proposed by Raymond’s E.A.L. Smith in the 1950’s. Since that time there have been many wave equation programs developed, including TTI, WEAP, TNOWAVE and the TAMWAVE available on this site. Most of these not only include the discrete mode necessary for proper modelling of the system but also convenience features such as a pile hammer database, a pile database, an estimate of the driving time and estimating the static resistance of the pile given basic soil properties and profile.
An important result of wave equation analysis is the socalled “bearing graph” output.
The wave equation returns its results as a function of the blow count of the hammer, i.e., the number of blows per unit length of pile penetration the hammer impacts during driving. There are three important results to consider:
 Ultimate Capacity of the pile. Although it can be estimated from soil properties, the actual ultimate capacity will vary from this estimate, significantly in some cases. During driving, if the blow count varies substantially from what was anticipated, this indicates that the actual ultimate capacity varies significantly from what was estimated. (One phenomenon that complicates this is pile setup. Especially pronounced in clays, with setup the ultimate capacity during driving is significantly lower than when the pile is in service. Frequently in this case a “restrike” is done after setup to insure the wave equation’s estimate is correct.)
 Maximum compressive stress in the pile. As is the case with the ultimate capacity, this increases in a “diminishing returns” fashion to approach a maximum value, i.e., the value of a semiinfinite pile.
 Maximum tension stress in the pile. This is generally experienced in the low blow count/resistance range, for reasons noted earlier. This is important to monitor in order to avoid pile damage.
The wave equation in this form can thus be used both during design of the system and during construction, and is a very powerful tool for driven piles.
Dynamic Pile Monitoring
Up to now we have dealt with either theory or a numerical simulation. But is it possible to apply wave mechanics to actually monitor the pile during driving? The answer is not only yes, but dynamic pile monitoring is a key component in the verification of pile design, both from the stress control standpoint and the estimate of final ultimate capacity.
We saw that the nature of the reflection can tell us something about the resistance of the pile at the toe. The same can be said both for resistance along the pile shaft and for defects/slack points in the pile itself, although the analysis is more complex.
Ideally, one should monitor the stress waves at several points along the pile. This was the way it was done with the important studies during the 1930’s by Glanville et.al. (1936). They used piezometers at the pile head, toe and at the midpoint, and photographed the results as they appeared on the oscilloscope.
However, the fact that piles reflect the waves that are generated during impact enables us to monitor the pile top only. Figure 13 shows strain gauges and an accelerometer mounted near the pile top. In this way we can monitor two things: the forcetime history via the straintime history and the theory of elasticity, and the accelerationtime history, which can be integrated to the velocitytime history.
Figure 14 shows a Pile Driving Analyser, into which the signals from these strain gauges and accelerometers are fed. The screen shows the data coming from the pile during driving. As the name implies, the Analyser also processes the data.
For a semiinfinite pile, Equation 9 says that these two histories should be identical. But since we are not impacting on a semiinfinite pile, the difference between the two can be educational. This is a key concept in dynamic pile monitoring.
CAPWAP
The most common method of interpreting the information obtained from dynamic pile monitoring in use today is CAPWAP (CAse Pile Wave Analysis Program.)
In CAPWAP, the forcetime and acceleration time data is monitored and fed into a wave equation model of the pile. As Figure 15 shows, the method is iterative. The model computes a forcetime curve at the pile head and compares it with the actual data, then readjusts the model parameters (spring constant, dampening, etc.) and their distribution along the pile until the two forcetime histories match.
Figure 16 shows such a matching process. The method used to match the two has varied over the years, but the principle has been the same. This process is done in real time. The reliability of the method is such that it can be used as a substitute for a static load test in many cases, because the dynamic and static components of the resistance are separated during the analysis. In addition to being used for driven piles, it also can be used for bored piles (drilled shafts, augercast piles) if special hammers are employed to generate the forcetime curve necessary for the analysis. (One necessary prerequisite for a successful CAPWAP analysis is that the shaft and toe resistance be fully mobilised during driving.)
Case Method
The success of CAPWAP has been based on the increasing computing power available in the field to process the data. (CAPWAP, and other dynamic monitoring, can also be remotely monitored as well.) But what if there were a “back of the envelope” type of analysis that could be used to estimate ultimate pile capacity based on the forcetime and velocitytime data? That is the concept behind the Case Method, which actually antedates CAPWAP. The Case Method is also useful to give an initial feel for the use of field data, and that is our primary interest in it.
The Case Method was derived from a closed form solution of the wave equation, albeit different from the one used to generate the graphs shown earlier. Implicit in the Case Method are three assumptions:
 The pile resistance is concentrated at the pile toe, as was the case with the closed form solutions above.
 The static toe resistance is completely plastic, as opposed to the purely elastic resistance modelled above. (Both the wave equation numerical analysis and CAPWAP assume an elastoplastic model for the static component of the resistance.
 The dynamic toe resistance is proportional to the velocity of the pile toe.
The basic equation for the Case Method is as follows:
Equation 25:
Where
RTL = total resistance of the pile (static and dynamic,) N
F1 = pile head force at the peak force of impact (or other time,) N
F2 = pile head force at a time 2L/c later than F1, N
V1 = pile head velocity at the peak force of impact (or other time,) N
V2 = pile head velocity at a time 2L/c later than F1, N
As is the case with any dynamic method of measuring pile capacity, it is necessary to have a method to remove the dynamic resistance of the pile. The dynamic resistance is defined as follows:
Equation 26:
Where
RD = dynamic resistance of the pile, N
J = Case Damping Constant, dimensionless
The static resistance is computed by subtracting Equation 26 from Equation 25, thus:
Equation 27:
Where RS = static resistance of the pile, N
The simplest way to illustrate the use of the Case Method is through an example.
Example of Case Method
A pile has an impedance of 381 kNsec/m. Considering the forcetime curves below, determine the Case Method ultimate capacity for the RSP and RMX methods.
The values for the data points are as follows:
 FT1 = 1486 kN
 FT2 = 819 kN
 VT1 = 3.93 m/sec (Z∙VT1 = (381)(3.93) = 1497.33 kN)
 VT2 = 1.07 m/sec (Z∙VT2 = (381)(1.07) = 407.67 kN)
There are three things that need to be noted in interpreting Figure 17.
 There are two curves, both at the pile top. The first “F” curve (solid line) is the forcetime history of the impact blow. The “V” curve (dashed line) is the velocitytime history. Generally speaking, the velocity history is multiplied by the pile impedance, as is the case here. This is not only to make the two quantities scale properly on one graph; as noted earlier, if the pile were semiinfinite, the two curves would be identical. This is in fact the case in the early portion of the impact; neither pile movement relative to the soil nor reflections from the shaft are a factor until later.
 Case Method results can be interpreted in several ways. The method shown in the graph is the RSP method, best used for piles with low displacements and high shaft resistances. The t1 for the RSP method is the first peak point in the forcetime curve; the time t2 is time 2L/c after that. The time t1 is not the same as the time t = 0 in the closed form solution, or the very beginning of impact.
 A Case damping constant J = 0.4 is assumed.
Substituting the values taken from the graph into Equation 27,
The first term is the total resistance RTL and the second is the dynamic resistance RD.
Another method of Case Method evaluation is RMX.
The values here are as follows:
 FT1 = 819 kN
 FT2 = 1486 kN
 VT1 = 1.92 m/sec (Z∙VT1 = (381)(1.92) = 731.52 kN)
 VT2 = 0 m/sec (Z∙VT2 = (381)(0) = 0 kN)
We now note the following about Figure 18:
The time t1 is now the peak initial force plus a time shift, generally 30 msec with the RMX method (Fellenius (2009).) The time t2 is still t1 + 2L/c. This time shift is to account for the delay caused by the elasticity of the soil. (One of the implicit assumptions of the Case Method is that the soil resistance is perfectly plastic.)
The RMX method is best for piles with large toe resistances and large displacement piles with the large toe quakes that accompany them. The quake of the soil is the distance from initial position of the soilpile interface at which the deformation changes from elastic to plastic, see variable “Q” in Figure 11. The toe quake is proportional to the size of the pile at the toe.
The Case damping constant for the RMX method is generally greater than the one used for RSP, typically by +0.2, and should be at least 0.4. In this case we will assume J = 0.7.
Substituting the values taken from the graph into Equation 27,
There are other methods by which to evaluate the Case Method. However, the “tricky part” of the Case Method is determining the Case damping constant.
Figure 19 lists “typical” values of Case damping constants. However, the reality is that the Case damping constant is a “jobspecific” quantity which can and will change with changes of soil, pile and even pile hammer. These require calibration, either with CAPWAP or theoretically with the wave equation program. The Case Method requires a great deal of experience and judgment in its application to actual pile driving situations.
Variations in ForceTime Curves
One more thing that should be noted with dynamic pile monitoring is that the nature of the results will vary with the pilesoil system. Some variations are shown in Figure 20.
The difference between the top two graphs is due to the different nature of the reflected wave from the toe. Minimal toe resistance causes a “freeend” opposite sign type of reflection resulting in a slightly negative pile top force and tensile stress at 2L/c. Maximum toe resistance results in a compressive wave reflection, with a very positive pile top force and compressive stress. Large shaft resistance “muddies” the results, as reflections are taking place all along the pile shaft.
Conclusion
Pile dynamics is a complex yet important subject for deep foundations. The ability to dynamically evaluate pile capacities and stresses is essential for proper installation and performance of driven pile foundations, and the testing can be applied to other deep foundations as well.
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