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An analytical study was undertaken to develop a set of closed form equations to make preliminary estimates of the pile loads, stresses, dynamic resistance and other important parameters in the installation of offshore piling. Solutions were developed for cushioned and cushionless hammers alike, along with illustrations of sample results and comparisons with field-correlated wave equation results.

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... Pile head response from impact of pile hammers using semi-infinite pile theory has been used since Parola (1970), and the theory is explained in Warrington (1997). Up until this time its use has primarily been with long steel piles, such as are used in offshore platforms and wind farms (Deeks and Randolph (1993); Warrington (1987Warrington ( , 2020.) Although some of these models include the use of hammers without a hammer cushion, none of them include a cushion between the driving accessory and the pile head. ...

... developed further by Warrington (1987); ...

... In Warrington (2020) the parameter ranges were based on Warrington (1987), namely 0.1 ≤ z ≤ 1.6 and 1 ≤ m ≤ 10. For the concrete piles it is reasonable to assume that one or both of these ranges will be different, and additionally k will have to be considered. ...

The application of semi-infinite pile theory to the behaviour of driven piles has been studied since Parola (1970). Most of the effort, however, has been concentrated on piles which do not require a cushion between the pile head and the pile driving accessory, such as steel piles. Concrete piles, on the other hand, are generally driven with this additional cushion. In this paper the same type of semi-infinite type of analysis is applied to this problem. Both the case of a rigid pile head and a pile head which responds without reflection from the pile are studied using both closed-form and numerical solutions. Two case histories are included which illustrate the application of the method, along with parametric studies of both pile head conditions.

... Warrington (1987) was a detailed analysis of the mechanics of impact pile driving using semiinfinite pile theory and pile impedance concepts. In this paper a review and expansion of the basic concepts set forth in this paper is performed. ...

... The first comprehensive attempt to combine semi-infinite pile theory with pile impedance-matching-and thus using it as a possible predictor of pile performance during driving-was Parola (1970). Warrington (1987) continued with that analysis and extended it to impact hammers without a hammer cushion. This work also attempted to use the results of semi-infinite pile theory to estimate the static capacity of the pile, although the realities of wave theory and resistance characteristics of actual piles hindered the progress of the work. ...

... Use of this did not end with Warrington (1987); Deeks and Randolph (1993) used an elegant closed-form solution to predict pile head performance during impact. Warrington (1997) also developed closed-form solutions for both pile head and general pile response to impact. ...

Warrington (1987) was a detailed analysis of the mechanics of impact pile driving using semi-infinite pile theory and pile impedance concepts. In this paper a review and expansion of the basic concepts set forth in this paper is performed. First, a review of the basic concepts and equations set forth in the original is first performed with revised notation. From here a different numerical method is used to analyze the equations and a parametric study of the results is performed, giving an illustration of the resulting trends. Finally three test cases, one of which came from the original paper, are done to compare the results of the present model to specific situations.

... Therefore, any potential advancements in the field of pile driving analysis lie in the use of the wave equation. There are also a number of methods for pile dynamic analysis that are based on semi-analytical approaches other than the method of characteristics: (1) solutions for piles of semi-infinite length (Parola 1970, Van Koten et al. 1980, Warrington 1987, Deeks 1992, Parker et al. 1996, (2) solutions using the method of images (Glanville et al. 1938, Hansen and Denver 1980, Uto et al. 1985, (3) solutions by Fourier series (Wang 1988, Espinoza 1991), (4) solutions by Laplace transforms (Zhou and Liang 1996). The main disadvantage of the semi-analytical methods, including the method of characteristics, is that they involve complex mathematics that hinder the implementation of realistic soil reaction models. ...

The general aim of the present research is to identify areas of improvement and propose changes in the current methodologies followed by INDOT for design of axially loaded piles, with special focus on the dynamic analysis of pile driving. Interviews with INDOT geotechnical engineers and private geotechnical consultants frequently involved in INDOT’s deep foundation projects provided information on the methods and software currently employed. It was found that geotechnical engineers rely on static unit soil resistance equations that were developed over twenty years ago and that have a relatively large degree of empiricism. Updated and improved static design equations recently proposed in the literature have not yet been implemented in practice. Pile design relies predominantly on SPT data; cone penetration testing is performed only occasionally. Dynamic analysis of pile driving in standard practice is performed using Smith-type soil reaction models. A comprehensive review of existing soil reaction models for 1-dimensional dynamic pile analysis is presented. This review allowed an assessment of the validity of existing models and identification of their limitations. New shaft and base reaction models are developed that overcome shortcomings of existing models and that are consistent with the physics and mechanics of pile driving. The proposed shaft reaction model consists of a soil disk representing the near field soil surrounding the pile shaft, a plastic slider-viscous dashpot system representing the thin shear band forming at the soil-pile interface located at the inner boundary of the soil disk, and far field- consistent boundaries placed at the outer boundary of the soil disk. The soil in the disk is assumed to follow a hyperbolic stress-strain law. The base reaction model consists of a nonlinear spring and a radiation dashpot connected in parallel. The nonlinear spring is formulated in a way that reproduces realistically the base load-settlement response under static conditions. The initial spring stiffness and the radiation dashpot take into account the effect of the high base embedment. Both shaft and base reaction models capture effectively soil nonlinearity, hysteretic damping, viscous damping, and radiation damping. The input parameters of the models consist of standard geotechnical parameters, thus reducing the level of empiricism in calculations to a minimum. Data collected during the driving of full-scale piles in the field and model piles in the laboratory are used for validating the proposed models.

This chapter presents the design, manufacturing, and performance of a single-acting electro-pneumatic pile hammer, suitable
for installing model piles in a laboratory calibration chamber. The performance of the hammer was evaluated by instrumenting
the various components of the hammer-pile system and monitoring critical parameters such as pressure, displacement, force,
and velocity. The hammer delivers a rated energy of 156 ft-lb with a 66% efficiency. The hammer can operate with the full
design energy at frequencies up to 1.2 Hz, and has been used successfully to model the pile driving process in the laboratory.
Sufficient details are provided so that readers can build similar hammers to suit their needs.

The drivability study of piling requires modelling of the hammer impact to calculate the input force wave at the pile head. This force wave can be computed numerically or by using simple analytical solutions. In this paper, analytical solutions are presented for hammer impact using a model which uses lumped masses for the ram and anvil, a spring for the anvil cushion, and another spring for the cap cushion on the top of a pile. The pile is modelled as a dashpot. The solutions account for the separation of the anvil mass from the pile cap cushion as well as the separation of the ram mass from the anvil cushion. The developed solutions are used to perform a parametric study to illustrate the influence of the pile cushion on the maximum force transmitted to the pile head.

Small piles used in research programs have gener- ally been installed by pushing or driving using slow mechanical drop hammers, because of the expense and technical difficulties associ- ated with manufacture of realistic small-scale pile driving hammers. However, pushing does not model inertial effects and slow driving does not model diffusive effects in the soil. It remains to be demon- strated whether these effects are important or not. To help clarify this issue, an electronically controlled, single-acting air hammer, with a rated energy of 211 J (156 ft · lb), and an operating frequency of up to 1.2 Hz, was designed and built. The hammer was used success- fully to drive 90 mm (3.5 in.) diameter piles into dense sand under a confining pressure of 138 kPa (20 psi). The behavior of the hammer was documented using detailed measurements of time-dependent ram movements, accelerations, and chamber pressures. This paper is concerned with the design and performance of the hammer.

This paper presents a series of case histories on pile driving in the Gulf of Mexico, demonstrating the value of a pile-drivability analysis to the engineer planning an offshore pile-driving operation. Actual pile-driving records are compared with the results of pile-drivability analyses.
Introduction
Considerable research has been done in the past concerning the dynamics of pile driving. Increasing demands of the present-day offshore oil industry have necessitated construction of larger platforms in deeper water requiring larger piles with greater penetrations. As this occurs, the driving of the penetrations. As this occurs, the driving of the piles becomes more critical to the overall design. piles becomes more critical to the overall design. Recent attention has been focused on pile driving in new exploration areas such as the North Sea, Gulf of Alaska, and offshore California. The older, more active producing areas in the Gulf of Mexico largely have been overlooked. One of the aims of this paper is to is to demonstrate the usefulness and accuracy of pile-drivability analyses in the Gulf of Mexico. pile-drivability analyses in the Gulf of Mexico. A pile-drivability analysis provides significant benefits during both the design and installation phases. Its primary purpose is to ensure proper and phases. Its primary purpose is to ensure proper and efficient pile installation in the field. The analysis accomplishes this by aiding in the selection of proper hammer-cushion combinations, pile wall thicknesses, and add-on lengths for the particular site. The predicted blow counts from the analysis may be quite predicted blow counts from the analysis may be quite useful during pile installation in assessing hammer performance and actual soil conditions. This performance and actual soil conditions. This assessment allows the field engineer to determine any changes in equipment necessary during the pile-driving operation.
Background
The method of analysis used for pile-drivability analyses is the one-dimensional wave equation, first proposed by Smith, and now generally used for proposed by Smith, and now generally used for dynamic analysis of pile driving. Later improvements were made by Samson et al., resulting in the Texas A and M U. Wave Equation Program. Further modifications in the last several years have resulted in more efficient versions of the program, but no analytical changes have been made in the program. The version of the program used for this study is the TIDYWAVE program (Aug. 1975).
Basic Technique
The model used for the one-dimensional wave equation idealizes the pile system as consisting of a ram, cushion, pile cap, pile, and surrounding soil. The physical and analytical model of a typical pile is shown in Fig. 1. The pile hammer and pile are modeled as a system of concentrated masses connected by springs representing the stiffness of the pile and cushion. The soil is modeled by a spring and dashpot in parallel, attached to each concentrated pile mass below the mudline and at the pile point. pile mass below the mudline and at the pile point. The hammer and cushion properties used in the study are given in Table 1. The values shown are based on those recommended by researchers at Texas A and M U. The factors of hammer efficiency and cushion stiffness have been found to be fairly typical. While the actual hammer used is noted on the driving record, the type of cushion usually is not known.
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