**Example**

We are designing a pile cap for a bridge. We already calculated to forces at the top of the footing (we have not designed the footing yet and do not know the footing dimensions or the forces at the bottom).

The factored forces at the footing top are given as Pu = 4212 kips, Vu longitudinal = 108.4 kips, Vu transverse = 112.4 kips, Mu Longitudinal = 7340 ft-kips, Mu transverse = 16480 ft-kips.

Pile capacity is given as φRn = 424 kips/EA. Also, we never want there would be any uplift for the piles.

**Step 1) Footing dimensions**

We have found the footing dimensions as (31’-4”)×(15’-9”)×(9’-0”) either by “Iteration” or “Optimization.” (Or we can say that these dimensions are from engineering judgments or senior engineers’ instinct!).

**Step 2) Forces at footing bottom**

Pu = 4212 kips + 1.25(31’-4”)(15’-9”){(0.150 kcf)(9’-0”) + (0.100kcf)(2’-0”)} = 5168 kips

M Long = (108.4 kips)(9’-0”) + 7340 ft-kips = 8316 ft-kips

M Trans = (112.4kips)(9’-0”) + 16480 ft-kips = 17492 ft-kips

**Step 3) Find minimum required pile EA**

Try 6×4 = 24 EA

Try transverse pile spacing x sps as 68”, which has been found from “Iteration” or “Optimization”.

Longitudinal pile spacing y sps is selected as 51”.

**Step 4)**

The pile cap dimension is b = (6EA-1)(68”) + 2(1’-6”) = 31’-4”, h = (4EA-1)(51”) + 2(1’-6”)’ = 15’-9”, which match with the assumed value. Otherwise, we have to “Iterate”.

Other than the pile cap design itself, are we done with the pile arrangement?

**What are “Iteration” and “Optimization”?**

Pile spacing vs Pile Cap Area (by program WSP USA-Pile)

The pile cap area is (31’-4”) × (15’-9”) = 493.5 ft^{2}.

This graph shows the pile cap area and corresponding y sps for each value of x sps. We can see that the trial value of x sps = 68” is the one to produce the minimum pile cap area; in other words, it is the “Optimum.” This value of x sps has never been assumed or guessed, but it was calculated by “Iteration.”

The graph shows that x sps = 5’-0” to 6’-6” range also gives a very economical pile cap area. Some engineers prefer equal spacing for both directions, and x sps = y sps = 5’-0” would be a good choice for them.

The engineer can decide pile spacings easily by this graph.

**Closure**

Some engineers (include me) are obsessed with mathematical solutions and sometimes forget what is important. In the given figure, the x sps range 60” to 78” makes practically no difference, and trying to find the absolute minimum makes little sense. The engineer should try to determine the best pile spacing in the given range, rather than concentrate on small numbers.

We can solve this problem using the “Optimization” theory. We can ask the computer to find x sps and y sps which would give the minimum area of the pile cap, while the maximum reaction is less than φRn = 424 kips/EA. This methodology will be discussed later.

Here only the output from the “Optimization” theory is given.

The “Optimum” pile cap dimension is x sps = 67.96”, y sps = 50.58”, and the pile cap area is 489.9 ft^{2}., which is close to 493.5 ft^{2}, we got from “Iteration”.

Editor: Edgar De Los Santos

e.santos@midasoft.com

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