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 Overview
"...your impressive MS thesis would be a dissertation in most places."
J. Don Murff, PhD, Texas A&M Unviersity
Retired geotechnical engineer, Exxon
When people speak of "using the wave equation" for their piling problems, they're seldom referring to a formula but to a numerical method and usually a computer program and/or an instrumentation technique in the case of in situ tests. Yet, when the wave equation is studied in elementary differential equations, some kind of "equation" or formula can be derived. How it looks obviously depends on the boundary and initial conditions of the problem.
Such results are "closed form solutions." While stress wave propagation in piles is a complex, non-linear problem, some useful information can be obtained from a closed form solution, if only to check the numerical methods for underlying problems.
An attempt at such a solution was undertaken by Don C. Warrington at the University of Tennessee at Chattanooga; his result is documented in the master's thesis Closed Form Solution of the Wave Equation for Piles. We have several items relating to this thesis:
(The image is of a deflection-time plane for a 50m long pile using the closed form solution.)
Thesis Slideshow
Abstract
This thesis details the research into the one-dimensional wave equation as applied to piles used in the support of structures for civil works and driven using impact equipment. Since the 1950's, numerical methods, both finite difference and finite element, have been used extensively for the analysis of piles during driving and are the most accepted method of analysis for the determination of driving stresses, dynamic and static resistance of piles. In this thesis the wave equation is solved in a relatively simple closed form without recourse to numerical methods. A review of past efforts to solve the wave equation in closed form is included. Problems that appear in previous related works are discussed and derived again, including the Prescott-Laura problem of the cable system stopped at one end and the solution of a hammer/cushion/cap/pile system for a semi-infinite pile. The latter is used to assist in the determination of a pile top force-time function that can be used to simulate the impact of the hammer on the pile. The basic equations, initial and boundary conditions are detailed, with the parameters adjusted to match actual soil dynamic behaviour while at the same time being a form convenient for closed form solution. To avoid difficulties due to spectral elements in the boundary conditions, a strain-based model of the radiation dampening in the pile toe was developed. The solution technique uses a Laplace transform of the semi-infinite pile problem for 0 < t < L/c (or for a time duration 0 < t < d, where d < L/c) and a Fourier series solution of the Sturm-Liouville problem thereafter. This solution is applied both to undamped and damped wave equations. The work includes comparison with existing numerical methods such as WEAP87, ANSYS, and Newmark’s method using Maple V.
Preface
Let us consider that if the ancients had kept to this deference of daring to add nothing to the knowledge transmitted to them and if their contemporaries had been as much opposed to accepting anything new, they would have deprived both themselves and their posterity of the fruit of their discoveries. Just as they used the discoveries handed down to them only as the means of making new ones, and that happy daring had opened the road for them to great achievements, so we should take the discoveries won for us by them in the same spirit, and following their example make these discoveries the means and not the end of our study, and thus by imitating the ancients try to surpass them.
This quotation, taken from the Preface to the Treatise on the Vacuum by the French scientist and Christian thinker Blaise Pascal, is as fitting way of beginning such a work as this as one can find. Although the wave equation itself has been investigated since the days of Bernoulli, the application of stress-wave theory to piles is relatively recent, going back to the early 1930's. Although it is an exaggeration to refer to those who first investigated these matters as "ancients," given the acceleration of the growth of knowledge and the application of technology the time between the first investigations of this problem and the present is in reality rather long.
In any investigation such as this the ideal goal is to come up with something truly novel, and many of such works emphasize their novelty to the denigration of those who have gone on before. While in some fields of endeavour this might be appropriate, in this case such sweeping novelty cannot be claimed. This work fits the mould as outlined by Pascal above: it takes the work that has been done before, advances it a step while realizing that there are many more steps before "perfection" is achieved.
The use of the analysis of stress waves in piles to determine everything from the performance of the hammer to the capacity of the pile is widespread today. Most of these methods use numerical methods for the analysis. The use of numerical methods came rather early in this history of stress wave application to piles, earlier in fact than the computer power really needed for practical application was readily available. Closed form solutions were either abandoned entirely or applied on a limited basis or in an ancillary way to other techniques.
The acceptance of these methods without a way to really compare them with some kind of "theoretical" result have left some involved in the analysis of pile driving uneasy as to the theoretical basis of the solutions employed. A great deal of work has been done to correlate the numerical models with field data. But are these adjustments being made to actual field phenomena or to underlying deficiencies in the methods we are using? The answer to this question is critical because without a solution to this problem we may be solving the wrong problem, and thus guaranteeing surprises in the future when a breakdown in our corrections is induced by unforeseen conditions. This is especially important in a geotechnical problem because the variables in a problem are generally complex and inadequately quantified.
It is for this reason that we are "backtracking" to a closed form solution in this thesis. In doing this we are forced to take a hard look at the underlying mathematical theory of the wave equation as it can be applied to piles. Putting together sound mathematical application with the basic physics of the problem is something that is frequently lacking (generally through no fault of the investigators) in works in this field. While in this thesis we have attempted to accomplish this, we have both applied mathematics in a different way and in the process acquired a new sense of humility because the complexity of the problem stretches the mathematics applied to the limit.
With these thoughts we proceed to our subject, realizing that we are indebted to those who have gone before us and hoping to be yet another link in the chain of knowledge and understanding to those who might come after. With regard to understanding, however, we close with a quotation from the great Jewish scholar Moses Maimonides, from his Guide to the Perplexed:
My son, so long as you are engaged in studying the Mathematical Sciences and Logic, you belong to those who go round about the palace in search of the gate...When you understand Physics, you have entered the hall; and when, after completing the study of Natural Philosophy, you master Metaphysics, you have entered the innermost court, and are with the king in the palace. You have attained the degree of the wise men, who include men of different grades of perfection. There are some who direct all their mind toward the attainment of perfection in Metaphysics, devote themselves entirely to God, exclude from their thought every other thing, and employ all their intellectual faculties in the study of the Universe, in order to derive therefrom a proof for the existence of God , and to learn in every possible way how God rules all things; they form the class of those who have entered the palace, namely the class of prophets.
Related Articles
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An overview of the development of the use of
one dimensional wave theory as applied to piling, from the work of Isaacs
to the present time. |
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This paper presents the application of the closed form solution for the damped wave equation to piles. The wave equation in numerical solution has been used for many years, generally without even a simple closed form counterpart. In this paper the closed form solution for the damped wave equation will first be stated and related to an actual pile driven into the soil. Following this is a discussion of the boundary conditions: the hammer at the pile top and the soil response at the pile toe. To avoid spectral components in the Fourier series eigenvalues and to preserve orthogonality, a new strain based soil model to simulate radiation dampening from the pile toe is proposed. A solution to this equation which involves the solution of the semi-infinite pile using Laplace transform for the first part of the impact followed by a Fourier series solution for the remainder. Comparison with numerical methods for a sample case is also presented. |
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This paper documents both the development of a closed form solution for the one-dimensional wave equation as it is applied to piles and its comparison to numerical solutions of the same problem. Wave mechanics have been used extensively in piles for many years but the solution of the wave equation has been almost exclusively a numerical one. The closed form solution used involves the solution of the semi-infinite pile solution immediately after impact and a Fourier series solution for times thereafter. This solution is compared with numerical solutions of different kinds for a given test case. The comparison shows variations between the closed form solution and the numerical methods that, although not egregious, are also not consistent from case to case. A wider variety of cases is needed to come to more general conclusions about the variations in these methods. |
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This paper is an analysis of pile toe plates
that are assumed to interact with elastic foundations. A solution to the
deflection and moment equations is derived and discovered to be in fact
made up of Bessel functions with complex arguments. A solution based on
the analysis of the series that make up the Bessel functions is performed.
The solution is presented in the form of charts based on dimensionless
parameters. A sample case is analysed and discussed. |
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This paper is a review and summary of the efforts made to develop a closed form solution of the wave equation as applied to driven piles. It discusses the early development of solutions and includes discussion of such topics as semi-infinite piles, solutions using both Fourier series and the method of images, and solutions specific to vibratory hammers. Results, advantages, and limitations of each of these methods are discussed. The rationale for the use of closed form solutions as opposed to the numerical ones is set forth. The paper concludes with a discussion of the possibilities of future research and sets forth the requirements for making this research successful. |
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A description of the basic principles of
energy transfer and efficiency ratings as they apply to pile driving
equipment and the piles they drive. |
(Implementation of an Analytic Solution for the Phenomenon of One-Dimensional Propagation of Waves in Piles and its Adaptation for Result Interpretation of Pile Integrity Test (PIT))
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Víctor Hugo Restrepo Botero
Pontifica Universidad Javeriana, Bogotá
We feature this paper because it references extensively our own Closed Form Solution of the Wave Equation for Piles, and because it takes the solution a step further with the use of Fast Fourier Transforms. In Spanish. We have this in two versions:
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We feature this thesis for several principal reasons.
First, it is unusual in that it deals with both soil dynamics and wave propagation in piles in the same place. These two subjects are obviously related but are not tied together as often as one would like.
Second, it is an extensive treatment on the subject of loading rate in pile load testing. The significance of loading rate in testing is important both for the proper interpretation of the results and in relating the determined load to actual loads on piles, all of which have some kind of load rate.
Third, the background section of the thesis deals with many of the same soil models that were used in our own thesis. The soil modelling described in that work was heavily influenced by the work of Alain Holeyman, who supervised Charue's thesis. The thesis also cites our own paper on the development of the ZWAVE wave equation analysis program, which used soil models that were a departure from Smith.
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Nicolas Charue
Catholic University of Louvain, Belgium
Soils, like several other materials, exhibit strong time-dependent behaviour which can be evidenced in terms of creep or strain-rate effects. The degree of this rheological behaviour varies with the type of soil, its structure, and with the stress history. This effect is exacerbated in pile load testing where the procedure duration tends to be shortened under increasing time pressures. The modelling needed to interpret the results therefore becomes more and more complex, including soil viscosity, wave radiation into the soil and other significant phenomena.
Within this framework, it would be interesting to study the influence of the loading rate on the load-displacement behaviour of the pile through the results of two testing procedures loading piles with variable duration (Static Loading Test (SLT) used as reference and Dynamic Load Test (DLT)). Based on these data issued from two national research programs organised by the Belgian Building Research Institute (BBRI), the objective of the research reported herein is to refine the rheological parameters characterizing the influence of the loading rate within the framework of a relevant pile/soil interaction model fed with dynamic measurements acquired during pile Dynamic Load Tests. The final goal is to predict and simulate the quasi-static pile load settlement curve.
After an overview of the loading rate effects in the literature through the experimental and modelling aspects, the dynamic data measured on field are analysed. It has been observed that some relationships exist between maximum quantities such as: energy transmitted to the pile, pile head velocity and force and the settlements (maximum and permanent) measured after a sequence of blows during a DLT event. These relationships are similar for the sandy and clayey sites and repeatable in time. It has been found that there are some critical quantities (correlated with the hammer drop height) from which a significant settlement of the pile is possible while the pile does not settle if these quantities are not exceeded.
The pile/soil interaction system is described by a non-linear mass/spring/dashpot system supposed to represent the pile and the soil, with constitutive relationships existing within and between them. These relationships account for the static and the dynamic or rheologic behaviour. A back-analysis process based on a matching procedure between measured and computed curves (force and velocity) allows one to describe the pile/soil interaction in terms of constitutive and rheologic parameters based on thedynamic measurements. After optimisation of the matching procedure, the parameters obtained are used to simulate the “static” load-settlement curve. The matching procedure is based on an automatic multi dimensional parameter perturbation analysis. Since the parameters influence the system response with a relative weight, they are sorted in order to optimise all the parameters by successively retrieving the most influential ones and working on the remaining ones.
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David M. Rempe
University of Illinois
1975
Research was conducted into the mechanics of pile driving with diesel hammers. The first step consisted of an investigation of the mechanical and operational details of diesel pile hammers. Then, a mathematical simulation of diesel hammer operation was developed for purposes of wave equation analysis, which is an analytical method for prediction of pile load capacity and driving stress on the basis of driving resistance (pile penetration per hammer-blow). Finally, the performance characteristics of diesel pile hammers and the factors affecting performance were studied. The details of diesel hammer design and operation are described. Differences in design and operation among the various types of diesel hammer are discussed as they relate to pile-driving effectiveness. Design features related to inclined operation and soft-ground operation are discussed. The mathematical model of the diesel hammer is described in detail, with emphasis on the simulation of diesel combustion, steel-on-steel impact, and interaction of hammer operation with the dynamic response of pile and soil. In wave equation analysis of diesel pile driving, the mathematical hammer model is combined with models of the pile and soil to produce a total simulation of the hammer-pile-soil system.
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Jerry F. Parola
University of Illinois
1970
Impact pile driving was studied by utilizing longitudinal wave propagation theory as ananalytical tool . Field data from pile driving jobs was used to establish the validity and usefulness of the analytical techniques developed herein.
The theoretical treatment of the dynamics of impact pile driving included an analysis of both the force generated at the head of the pile and the response of the pile tip to a generated force pulse. A model consisting of a hammer system operating on the head of an infinitely long pile was used to determine both the pile force pulse and the transmitted energy. The model was used to make a dimensionless parameter study of the factors influencing force and energy. The driver system consisted of concentrated masses for both the ram and the drivehead and an energy absorbing spring (both linear and nonlinear) for the hammer cushion.
Soil and pile responses were investigated with respect to an arbitrary force pulse in order to assess the variables controlling pile penetration and load capacity. Special emphasis is placed on soil and pile response a t the pile tip; the soil model includes viscous damping, mass and an elastic-plastic spring.
Characteristics of the hammer-pile-soil system as a whole are summarized. Theoretical results using wave propagation theory are compared with both case histories and commonly used dynamic formulas. Correlation of wave analyses and field case histories are used to support the conclusion that wave propagation theory is the proper theoretical tool for pile driving analysis.
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This dissertation was co-directed by Spencer J. Buchanan, Distinguished Professor of Soil Mechanics and Foundation Engineering at Texas A&M and before that founder and Chief of the Soil Mechanics Division of the U.S. Army Waterways Experiment Station. The Spencer Buchanan Lecture at Texas A&M, an important lecture in geotechnical engineering, is named in his honour.
John V. Perry taught Mechanical Engineering at Texas A&M for many years. On a lighter note, his department head, C.M. Simmang (who signed off on the dissertation,) was commenting to his class on a visit by the late President Gerald Ford to San Antonio in 1976. Shaking his head in disbelief, he said, "At least I had enough sense to shuck the tamale before I ate it."
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John Vivian Perry, Jr.
Texas A&M University
August 1963
This research was undertaken to determine the amount and extent of soil motions under vibrating foundations. The test soil was standard 20-30 Ottawa sand, ASTM C-190, that was contained in a one-meter cubical box. A force generator was mounted above the soil and applied dynamic loads to a circular footing. These were harmonic forces and were applied at frequencies between five and fifty cycles per second.
Three hundred and sixty-seven test runs were recorded on an electromagnetic oscillograph from signals generated by an acceIerometer buried in the soil. This acceIerometer was located at various depths beneath the center of a footing and, at other times, it was located beneath and offcenter. Other variables were the footings which had different diameters and masses.
Three empirical equations were developed from the test results using dimensional analysis. These equations were for maximum values of acceleration, velocity and displacement, respectively.
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This work is a classic for soil dynamics in general and the response of soil to the vibration of foundations in particular. Lysmer's simplification of the response equation was a major step forward in the rational analysis of this phenomenon.
Lysmer's Analogue--which reduced the soil response of a rigid circular foundation to a single degree of freedom spring-dashpot system--also has found application in pile toe response to pile driving, as was discussed in Closed Form Solution of the Wave Equation for Piles.
John Lysmer was for many years a Professor of Civil Engineering at the University of California at Berkeley. He passed away in 1999.
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John Lysmer
U.S. Army Corps of Engineers Contract Report 3-115
University of Michigan
June 1965
This investigation includes a theoretical solution for a rigid footing, resting on an elastic half-space, which is subjected to steady-state vertical oscillation. It is shown how this steady-state solution can be used to describe the response of the footing to a transient pulse-type vertical loading.
After establishing the theoretical solution, and evaluating the approximations required for its development, it is further demonstrated that the theory permits evaluation of quantities which may represent spring constants and damping factors for use in the usual theory for vertical motion of a damped-one-degree-of freedom system. The agreement between the simple theory and elastic half-space theory is well within the limit required for engineering solutions.
The results of the study provide information from which the elastic dynamic response of rigid footings subjected to transient vertical loads may be evaluated. By taking advantage of such standard procedures as the phase-plane method, the dynamic response of footings may still be estimated even if the stresses in the soil extend into the inelastic range. A detailed discussion of the application of this method to inelastic settlements of vertically loaded footings will be presented in a subsequent report.
Finally, the theoretical developments included in this report for vertical oscillations may serve as a guide to develop similar theoretical evaluations of the dynamic response of rigid footings in other uncoupled modes of oscillation.
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