Pile groups: FEM vs. Math(CAD)

One of the regularly returning structural elements is the so called pile cap, plainly put a huge block of concrete with multiple foundation piles to transfer loads from (a column in) a building to the earth underneath.

Start of pile cap reinforcement,
with the pile heads visible

Final pile cap reinforcement, with the connecting column reinforcement.
All it needs now is a bit of liquid concrete (just a few cubic meters :-P)

These piles have a finite stiffness, whereas the stiffness of the concrete block can be considered (close to) inifinite most of the times. Due to construction tolerances or obstructions in de ground underneath a building, these pile groups are not necessary symmetrical in regard to the transferred load. Therefore stiffness and position of the piles determine the load distribution to the piles.

With large pile caps, nowadays normal practice would be to model the pile cap in a FEM (Finite Element Model) software package and calculate it there. This can be quite time consuming, so for design purposes it would be nice to have an easy calculation sheet that does the same.

Just to demonstrate, I've made a FEM model of a pile cap with some eccentricity in pile grouping and loading, with reaction forces resulting from the calculation of this model:


I was working on just such a sheet to calculate stiffness and load distribution on a foundation block. I imagined writing loads on and piles under the foundation blocks in arrays and then processing those very efficiently to get load- & pile group centre, calculate loads, etc. For this one I chose to investigate usage of tables in MathCAD.

Defining sets of data with tables

The tables work out nice! And they make it quite easy to execute aforementioned operations on the entire table like determining the load- and pile group center. Using that data I want to calculate the location of the resulting force (from the left table) and the center of the pile group (right table). This is done by summation of the distance of a load or pile mutiplied by respectively the load or the stiffness. In MathCAD these are simple mathematical operators, using table row references:

The functions are elegant, but are at the same time taking up a lot of space on the sheet when defining them one by one. In programming this is usually tackled by defining a function, so why not introduce this to MathCAD as well? It turned out to be easier than expected:

This has two major advantages: first, I can use one function to calculate both the resulting load X&Y coordinates AND the X&Y coordinates of the pile group. Second, I can define the function at the start of the sheet, even before creating the tables above. That way, the function can be used everywhere in the sheet. So I put the function outside of the print area at the top of the file, and I can calculate directly following the tables, very compact:

One thing left: the operation leaves an 1x1 matrix variable (the brackets around the resulting values), which is not what I want. A small adjustment of the function (adding an index 0) will result in a single variable as the result of the function:

 

Next step is calculating the distance from each load or pile to the group centre (left function below), and the total load on the pile group (right). Again I introduce functions. They're quite simple really, but make the sheet very scalable:

These functions help determine the overall pile cap make-up: load and pile centres (res subscripts), distances to load and pile group centres (z subscripts). From there the pile group properties and external forces on the group can be determined:

The content of Xp.z and Yp.z is not shown above. The zc function operates on a Matrix, resulting in another matrix:

I will not go into too much detail on the structural mechanics behind the stiffness of the pile group (Ix, Iy and Ixy); what it boils down to is that these properties determine the load distribution to the piles. Using these values, we can finally determine the load distribution and vertical deformations of the piles!

Resulting in:

Comparing these with the FEM calculation, an (almost) exact match! :-)

This pile cap clearly needs some added stiffness :-) To demonstrate that it also takes into account the stiffness of the supports, The stiffness of the first support is divided in half: k = 50 MN/m1. Also, a sixth pile is introduced with a stiffness of 200 MN/m1. Comparing MathCAD calculation with the sheet:

Advantage of a sheet like this one, is that it can be scaled to almost every size of pile (or anchor) group imaginable. As long as the transferring medium can be considered about infinitely stiff, of course...

One downside though: I have failed at making a viable pile cap. But hey, as long as it served a purpose!