# The Limitations of Dynamic Formulae

In an earlier post Vulcan’s advice on the limitations of dynamic formulae–specifically referencing the Engineering News Formula–was set forth.  That advice appeared in Vulcan’s literature from 1906 to 1931.  Later in the 1930’s Vulcan changed the way its literature was organised, and in the process Vulcan added new verbiage (a lot of it, really) to its literature.  That included a new “disclaimer” on dynamic formulae.   The version in Vulcan Bulletin 68 (the first for the Warrington-Vulcan hammers) is shown here, and appeared in Vulcan literature into the 1960’s:

It is not possible to devise a simple, yet accurate, formula for determining the load bearing capacity of a driven pile and incorporate, therein, allowances for all the modifying conditions that arise.  The piles may be long or short, light or heavy, of different cross-sectional areas and materials, and may have been driven with or without a cushion block.  Soils very greatly.  Diverse styles of pile hammers are employed for driving; some may have, in comparison to the total weight of hammer, or with respect to the weight of the pile that is to be driven, a relatively light ram, others a relatively heavy ram, and may strike few or many blows per minute at widely different velocities.

A simple formula cannot take into account the existing ratio of the weight of the striking part (or ram) to the weight of the pile, though both this and the velocity at the moment of impact are factors which greatly affect the efficiency of the blow and, consequently, the bearing capacity of the pile.  The effectiveness of the cushioning, elasticity of the pile and quake of the soil are other variable factors of vital importance not possible of inclusion in a simple formula.

A manufacturer of dynamic equipment such as Vulcan, which interacted with end users in many locations, was in a better position to grasp the complexities of dynamic pile driving than many foundation engineers of the time.  The 1930’s, however, was the decade when change began, when Isaacs and others began to explore the use of wave mechanics to analyze the performance of driven piles.  The pushback from that was intense, but ultimately E.A.L. Smith at Raymond took the lead and developed the wave equation, which would address the issues that Vulcan had brought up in a qualitative way.