The whole point of impact pile driving is for a moving mass to be stopped “suddenly” and in the process generate a high peak force impulse which moves the pile into the soil. Pile dynamics people tend to emphasise the force on the pile head, while hammer manufacturers tend to emphasise the peak force of the ram. In this piece we’ll do the latter, as knowing this force is important for proper design of the components.

In the context of Vulcan, we’ll look at this in two ways: the “old Vulcan way” which was based on Newtonian impact mechanics, and the “new Vulcan way” which was developed at Vulcan in the last half of the 1980’s and has been revived and expanded on recently in the posts OTC 5395 Revisited: Analysis of Cushioned Pile Hammers and Concrete Pile Head Response to Impact.

## The “Old Vulcan Way”

In the post Billiards and Pile Driving: Newtonian Impact Mechanics and Dynamic Formulae we discussed the direct application of particle dynamics to pile driving. Dating back to at least the 1950’s, Vulcan used a very primitive method to estimate the force of the ram and by extension the maximum deceleration of the ram during impact. Ignoring the effects of the pile, and equating the momentum at impact to the impulse during impact yields

(1)

where

- M
_{1}= mass of ram, slugs - V
_{1}= velocity of ram at impact, ft/sec - F = force of ram during impact, lbs.
- t = time (of impact,) seconds

Vulcan made two assumptions. The first was that the impact force was constant, which allows the equation to be written as

(2)

If we use the weight of the ram rather than its mass, we can write this equation as

(3)

where

- W
_{1}= weight of ram, lbs. - g
_{c}= acceleration due to gravity = 32.1939 ft/sec^{2}

from which

(4)

We could also directly compute the acceleration in g’s by dividing through by the weight of the ram, thus

(5)

where g_{max} = maximum deceleration of the ram, g’s.

The second assumption was that the impact time duration was only 1 msec. To see how this worked, let us consider the following specifications for offshore hammers from the late 1970’s:

Let us further consider the 560. The specifications show that the hammer has a ram weight of 62,500 lbs. and a net striking velocity of 16.7 ft/sec. Substituting these and the impact time assumption into Equation (4) yields a ram force of 32,440,892 lbs., which is very close to the specifications. Substituting the same information into Equation (5) yields a deceleration of 519 g’s.

Fortunately few people paid attention to this part of the specifications, and the shortcomings of this method will be explained in comparison with the “new” method described below.

## The “New Vulcan Way”

In 1987 Vulcan personnel presented a paper entitled “A Proposal for a Simplified Model for the Determination of Dynamic Loads and Stresses During Pile Driving” at the Offshore Technology Conference. The application of wave mechanics to pile driving changed the way we look at the interaction of hammer, pile and soil during impact. The problem for manufacturers is complicated by the multiplicity of piles and soil stratigraphies that hammers drive piles into. A simplification of that can be done by applying semi-infinite pile theory, which is explained here.

For a cushioned hammer, if we assume the “base” (in this case the cap) is rigid, the simplest way to depict the system is as a one degree of freedom system with an inextensible spring. The force-time curve is a sinusoidal impulse that begins at zero, reaches its maximum at the maximum deflection of the cushion, and returns all of the energy to the ram back at the zero deflection point. The equations of motion for this are discussed in OTC 5395 Revisited: Analysis of Cushioned Pile Hammers. The maximum deceleration of the ram in g’s is given in the 1987 paper as

(6)

where

- β = maximum deceleration of the ram, g’s
- e = efficiency of the hammer from ideal
- S
_{o}= stroke of the hammer, ft. (paper assumes a single-acting hammer, would be an equivalent stroke for one with downward assist) - k = stiffness of the cushion, lb/ft

The length of time from initial contact of the ram with the cushion to its return to flight is referred to as the ram period, or

(7)

The 560 case in the original paper was re-analysed in OTC 5395 Revisited: Analysis of Cushioned Pile Hammers. The basic parameters of this study are given in the table below.

Rated Striking Energy, ft-lbs | 312500 |

Weight of Striking Parts, lbs. | 62500 |

Equivalent Stroke, ft. | 5 |

Efficiency, percent | 75 |

Net Striking Energy, ft-lbs | 234375 |

Impact Velocity of Ram, ft/sec | 15.53 |

Cushion Stiffness, kips/ft | 306796.2 |

Ram Frequency, rad/sec | 397.41 |

Maximum Theoretical Ram Deceleration, g’s | 191.87 |

Ideal Impact Force, kips | 5996.1 |

Hammer Impedance, lb-sec/ft | 771992.6 |

Ram-Cap Mass Ratio | 1.95 |

System Length, in. | 0.469 |

The ram frequency above translates into a ram period of 7.9 msec. All of this shows the two weaknesses of Vulcan’s old method from a theoretical standpoint. The first is that the waveform is not a square wave. The second is that the impact time of 1 msec is unrealistic. (It should be noted that the efficiency given here is different; using the same calculations above, the maximum deceleration in g’s is 483.)

An interesting comparison between square and sinusoidal waves is shown at the top of the page, which comes from Closed Form Solution of the Wave Equation for Piles. Because of the mathematical solution under consideration, it was necessary to “convert” the sinusoidally based solution from that model to a square wave. The duration of the square wave was set at 9.636 msec.

Further attenuation of the ram force takes places with the compression of the pile head, even without the general movement of the pile. OTC 5395 Revisited: Analysis of Cushioned Pile Hammers discusses this in detail. How much attenuation takes place depends upon the impedance of the pile, which in turn is a function of the cross-sectional area and material properties of the pile, relative to the hammer. For the cases considered, the maximum ram deceleration varied from 114 (for an impedance ratio of 0.2) to 138 (for an impedance ratio of 1.5.)

The situation is more complex for concrete piles, and this is discussed in Concrete Pile Head Response to Impact. In general concrete piles are “easier” on hammers than steel ones, but there are certainly exceptions. This is due to a) the need to limit stresses in concrete more than steel and b) the existence of the double cushion, which however can be challenging for driving accessories.

## Conclusion

Like pile head forces, hammer forces and decelerations can be estimated with a reasonable degree of accuracy, but to do so requires some understanding of the mechanics of the ram/driving accessory/pile system, and a realisation that no model such as the one presented here can be expected to anticipate all situations in the field.