Most people involved in the pile driving business know that, when you drive piles on a batter (angle) the energy from the hammer is reduced. But how much? This post attempts to answer that question, and show how Vulcan dealt with the issue.

Let’s start with single-acting hammers like the Warrington-Vulcan hammers: the rated striking energy is obviously

(1)

where

- E
_{r}= rated striking energy, ft-lbs - W
_{s}= ram weight, lbs. - s = stroke, ft.

When a hammer is put on a batter, its energy is reduced by two factors:

- The reduction in the fall through the gravity field. This is referred to as the “cosine effect” because the rated striking energy is multiplied by the cosine of the batter angle.
- The subtraction of the drag due to mechanical friction between the ram and the columns/ram tube. The drag is a product of the coefficient of friction of the ram riding on the columns/ram tube and the force of the ram on the frame, which should probably be referred to as the “sine effect.”

The effect of those two is shown by the equation

(2)

where

- E
_{rd}= reduced rated striking energy on a batter, ft-lbs - B = batter angle expressed as a fraction. Thus, a batter of 1:3 has B = 1/3, 1:4 has B = 1/4, etc. To use this the numerator has to be 1, thus a 9:12 batter has B = 1/1.333
- f = coefficient of friction

Using trigonometric identities eliminates the sines and cosines and reduces them to radicals.

Dividing Equation (2) by Equation (1) yields

(3)

where r_{diff} = reduction ratio in energy.

So how did Vulcan deal with this problem? The only place where it did was in its Data Manual, which has been incorporated into the vulcanhammer.info Guide to Pile Driving Equipment. The relevant page for single acting hammers is shown below.

There are two significant differences between this formula and the one we derived. The first is that Vulcan incorporated “an assumed friction between leaders and hammer of 10%.” (What the friction between the leaders and the hammer has to do with this is hard to know, although keeping leaders greased during operation is important.) In any case this translates into a coefficient of friction f = 0.1. The second is that Vulcan’s formula expresses the portion of “plumb energy” as a percentage.

For the example case with a Vulcan #1 Hammer, if we substitute f = 0.1 and B = 1/3 into Equation (3), we get r_{diff} = 0.917, which is identical to Vulcan’s result.

The situation with differential acting hammers (or any hammer with downward assist) is more complicated. The rated striking energy for a plumb Super-Vulcan hammer is

(4)

where

- A
_{sp}= area of small piston sq.in. - p
_{inlet}= inlet pressure at the hammer, psi

The reduced energy on a batter is thus:

(5)

Dividing the two equations as before yields

(6)

Now we have a problem; the reduction depends upon the nature of the downward assist, or more specifically the relationship between the free drop energy and the energy derived from the downward assist. If we define f_{rat} = (Force due to downward assist)/(Force due to free drop) = (A_{sp} p_{inlet})/(W_{s}) without effect of a batter, we obtain

(7)

For a single-acting hammer, f_{rat} = 0, and Equation (7) reduces to Equation (3) as it should.

Vulcan had a formula for this too, as can be seen below.

For a Vulcan 50C Hammer, f_{rat} = (55.917)(120)/(5000) = 1.341. Substituting this and the variables from the previous problem yields r_{diff} = 0.965, or a batter rated energy of 14,571, which is below Vulcan’s calculation.

Some comments are in order.

- The coefficient of friction used above in both calculations is possible but probably unrealistic in many cases. One key advantage of Vulcan hammers is that they will run with minimal maintenance, but that has a price, and one of them is the efficiency of the hammer. Coefficients of friction up to 0.5 and beyond are possible. For example, if we increase the coefficient of friction to 0.3, the values of r
_{diff}for the above calculations decrease to 0.854 and 0.938 respectively, which is substantial. This underscores the importance of proper cylinder and column lubrication as described in Vulcan Offshore Tip #23: Lubrication and in the Field Service Manuals, contained in the vulcanhammer.info Guide to Pile Driving Equipment. - The f
_{rat}concept shown above applies to hammers with relatively constant downward assist forces. It needs to be modified with hammers which use a gas buffer such as the old Link Belt/ICE diesels or the IHC hammer. - The calculations above do not take into account losses due to fluid back pressure or losses due to packing or piston ring friction. These need to be considered when estimating the total mechanical efficiency of a pile hammer when run plumb or batter. The matter of back pressure is discussed in The Valve Loss Study.
- These calculations do not consider the issue of rebound and its effects with differential hammers.