# Other Monographs on the Wave Equation

### Analytical Interpretation of Pile Installation and Axial Performance

#### Hudson Matlock, Ignatius Lam and Lino Cheang

A program of combined experiment and analytical studies is presently underway with the purpose of extending present understanding of the axial behaviour of pile foundations. The analytical developments are designed for back fitting and correlating the experimental results and for extrapolation to prototype designs. Emphasis therefore is placed on versatile and general-purpose computation tools which will permit examination of a wide variety of soil modelling concepts.

### Comparison of Numerical Methods to Closed Form Solution for Wave Equation Analysis of Piling

#### Don C. Warrington and Robert H. Wynn, University of Tennessee at ChattanoogaNovember 2000

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.

### Comments on One-Dimensional Wave Mechanics

#### Dhanvin Mehta

The classical wave theory, which describes the propagation stress waves in the rods, forms the underlying basis of pile driving theory and pile driving computer programs. In this respect, piles may be considered as rods and the same analogy can be extended to the other parts of the driving system. The purpose of this study is to describe essential differences in the development, modelling as well as basic ideas behind the computer program popularly called ‘WEAP‘ (Wave Equation Analysis of Piles) which was developed for the Federal Highway Department, and the algorithm based on wave equation theory, extended and refined realistically t o include complications due to presence of discontinuities in pile cross section, skin friction as well as internal damping. Only air steam hammers are considered as a part of the latter approach mainly because of the simplified model adopted, and for the ease in the program development. The conflicts associated with any other hammer system during modelling, using wave theory approach are self evident, and needs no further elaboration .

### Dynamic Behaviour of Battered Piles

#### Mohammad Ettouney and Jeffrey Janover

The one-dimensional wave equation is used in many forms. Unfortunately it is not able to take into consideration the lateral effects in battered piles. Battered piles are studied in dynamic situations. It is concluded that the dynamic effects of pile batter exceed those of static effects.

### The Effects of Material Damping on Wave Equation Analysis of Pile-Driving

#### Rick Corder, George Cozart and Jim FieldTexas A&M University

A study on the effects of material damping in wave equation analysis of pile driving, with modified factors to properly take this into account.

### Numerical Approximations in Pile-Driving Analysis

#### R.O. Davis and P.J. PhelanInternational Journal for Numerical and Analytical Methods in Geomechanics, 1978

A finite-difference scheme widely used in analysis of pile driving is examined. The scheme is shown to yield peculiar results in simple symmetric impact problems and to inherently possesses an imprecise definition of energy balance. A modified difference scheme is suggested and comparisons of the two methods are shown.

### An optimisation method for pile driving analysis

#### Jeremy Dolwin and Trevor J. PoskittQueen Mary College, London

A numerical method has been developed for analysing field data from piles as they are driven. The basis is a “least squares” technique which is used in conjunction with a non-linear optimisation procedure. It enables the measured stress time curve to be fitted by a wave equation analysis by calculating the values of hammer impact velocity, cushion stiffness, coefficient of restitution, soil quake, damping and skin friction such that the difference between the observed and predicted stress-time curves is a minimum. Good estimates of these parameters are of considerable practical value because they enable a comparison to be made between the data assumed at the design stage and that actually observed during installation. The acceleration-time curve produced by the optimised parameters is then compared with the measured acceleration curve. This provides a completely independent check on the validity of the calculated parameters. Since acceleration is a very sensitive variable in the computation, this is the most stringent test of the calculated values which can be devised. Finally, two worked examples are presented.

### Parameters for Friction Piles in Marine Soils

#### F. Baguelin; R. Frank and J.F. Jezequel

The aim of this paper is to show how self-boring pressure meter parameters can be used to predict the load-displacement curves of friction piles, by means of simple numerical analyses : either by load transfer functions analysis (section I) or by finite element analysis assuming a linear elastic behaviour for the soil mass (section II). In section III, the results of such theoretical analyses are compared with the experimental results of full-scale pulling out tests performed on two tubular piles driven in marine soils.

### Pile Driveability is Unpredictable in Sand or Silt Foundation Strata

#### Mari Mes, J. Ray McDermottPetroleum Engineer, October 1976

On a North Sea project, it became evident that the drivability estimates were inadequate to properly predict the depth of penetration of the pile. This paper proposes changes in the modelling scheme to improve the prediction of drivability.

### Sensitivity and accuracy of the pile wave equation

#### G.E. Ramey and A.P Hudgins

The dynamic wave equation provides a means of evaluating pile capacity that is mathematically well-founded and probably provides the most realistic model available for depicting actual behaviour of the hammer-pile-soil system. Numerical integration of this equation, with the aid of a digital computer, appears to be the most rational analytical means of evaluating pile capacity. A computer program solution of the wave equation was utilised in the investigation being reported to adjudge a) the sensitivity of the program-generated P-n curves to the program input soil parameters and b) the accuracy of the program in predicting pile capacity.

### Soil Strain Rate Effects on Axial Pile Capacity

#### R.G. Bea

Soil strain rates are known to have important influences on the load and deformation response of axially loaded piles embedded in cohesive soils . Generally, high rates of loading result in increased load resistance and stiffness. Low rates of loading result in decreased load resistance and stiffness . In this paper, results from recent deep-penetration pile load tests in which rapid rates of loading were applied, are used to describe strain rate effects. These results are compared with those from laboratory soil tests employing high rates of strain. A viscous damping coefficient is derived from these results . The viscous damping coefficient is utilized in a recently developed computer code (INTRA) intended for analyses of dynamic soil-structure interactions. The code and damping coefficients a r e used in a study of the dynamic response characteristics of a pile beneath a 1000 ft water depth platform located in the Gulf of Mexico. Based on results from an empirical approach, and the analytical approach discussed in this paper, substantial increases are found in the load resistance of the pile foundation subjected to typical combinations of static and dynamic loadings.

### Three-Dimensional Analysis of Pile Drivability

#### I.M. Smith, University of Manchester, and Y.K Chow, Fugro Ltd.

Pile drivability is usually assessed, in the offshore industry as well as elsewhere, on the basis of calculations which solve the one-dimensional wave equation. Clearly this is an approximation to the real situation, in which the driving process induces stress waves in the soil surrounding the pile. This paper examines the validity of the one dimensional approximation, by comparing it with a three-dimensional (axisymmetric) one. Both one- and three-dimensional idealisations are of the finite element type. In addition to the long-established use of wave equation predictions for drivability, recent advances in pile instrumentation during driving have led to the use of the one-dimensional equation as a means of analysing driving records with a view to predicting static capacity. In this sort of calculation, a few passages of the stress wave along the pile are analysed. In this paper, one-dimensional and three-dimensional models are compared in this context also. Finally, offshore piles are usually driven as open-ended hollow pipes and controversy exists as to how to treat the soil “plug” in one dimensional analyses. The ability of the three-dimensional analyses to shed light on this problem is briefly discussed.

### Wave Equation Analysis and Pile Driving Analyser for Driven Piles: 18th Floor Office Building Jakarta Case

#### Budijanto Widjaja

In many cases, the actual capacity of piles can be gained from static and dynamic pile tests. And this case is using dynamic pile test such as Pile Driving Analyser (PDA) for driven piles. In other hand, beside the bearing capacity of pile, the type of hammer with its weight, its drop, and soil stratification give responds. The responds vary and also give the information of pile’s displacement, pile’s tension forces, or pile’s tensile forces. This information is useful in pile’s performance and pile’s integrity. This paper examines the case of 18th floor office building in Jakarta. In this paper, both the effects of driving and the wave propagation in piles are analysed based on the kentledge system loading test (static test) and PDA. From the wave equation analysis which is proposed by Smith (1960), the geotechnical engineers can determine the dynamic capacity of pile and compare to the results of static test and PDA. The main problem is the spun pile had cracks along the pile from the head when conducting the static test in the certain degree of design load. Initially, the pile has a good result from Pile Integrity Testing (PIT). With combination of static test and PDA, bearing capacity graph for spun pile (P182) is generated. It is shown that final set is 0.3 mm and this value is close enough to driving record (0.2 mm). The cracks of the pile head during static load test may be caused by overstress after pile driving. It is modelled by wave equation analysis and shown that the damage of pile is mainly governed by compression force. The ultimate capacity of pile is 401.0 ton from static test. The wave equation analysis gives a conservative ultimate capacity value as 350.0 ton.

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