Onions

I am a long-time fan of onions. Raw, baked, in my salad or on my burger! But ... also onions in structural engineering, a.k.a. the concrete M-N-capacity curves for reinforced concrete sections:

My first introduction with the Onion was the in the Concrete Section Analysis (CSA) module of ESA Prima Win, the predecessor of the FEM software package SCIA Engineer. Of course there were and are more similar tools, but what has always been lacking in my opinion, was a way to do a "quick and dirty" design check on multiple combinations of normal forces (N) and bending moments (M).

Up until a few months ago, I myself mostly used tools like Technosoft's (Dutch-only software) "Kolomwapening" (Concrete column reinforcement) or "MN-Kappa" (for general concrete sections), or previously mentioned ESA (until ESA Prima WIN stopped to exist and the functionality of CSA ceased to exist). I even used IDEA's Statica once or twice the last few years.

The problem with those tools is that they are designed to (extensively) check sections: large numbers of options and settings in the software and not always the possibility to check a number of section force combinations at once.

So I first started to develop an easy Excel sheet to create a concrete "onion", based on some work of a colleague of mine at my previous employer (still a big fan, Sander!) His sheet of course was way too difficult for me, with complex VBA scripting, self-programmed Newton-Raphson iterations, crack width calculation, prestressing, etc. But it was a good starting point for my next exercise: how to create

The Onion

The basis in Excel was quite rudimentary: a fixed set of calculation lines, with varying strains on each line following the graph outline, calculating internal section forces and eventually the external section forces.

The general idea is to vary the strains in the section as follows:

(0) Full (plastic) pressure
(1) To full (plastic) tension in rebar side 1
(2) To full (plastic) tension in  rebar side 2
(3) To full (plastic) pressure in concrete outer fiber side 1
(4) To full (plastic) pressure in concrete outer fiber side 2

This is quite easy in Excel for one type of section (rectangular). The sheet below is the "background" calculation sheet:

Just add a presentable front sheet, with input options, capacity graph and the possibility to plot sets of M-N data, and you've got yourself a nice design sheet for ultimate limit capacities.

The result is also very robust (no VBA programming!), but unfortunately very limited in use (only rectangular sections, the accuracy of the graph is limited/fixed, etc.)

So I started to think on moving to MathCAD. There were 2 major steps to consider first:

1. How to (automatically) calculate internal and external forces
2. How to incorporate the strain flow described above and combine it with the calculations of internal forces

The first step 

screamed "FUNCTIONS!", see also my previous blog post about pile groups. So I started to define calculation of steel and concrete stresses based on strains, to internal forces, to external section forces. I've shown the used functions below, for those who are interested:

The result is fun, and can be easily visualized by plotting the functions, e.g. stresses and strains over the height of the section, with the corresponding external section forces:

The second step

was a bit harder. How to plot a graph in MathCAD with M-N relation, where N is not a function of M, or vice versa? So I tried a "step-wise" graph: Just define steps 0 to for strain (a function with 4 sections, one for each trajectory on the onion contour), make the strain follow the onion contour step by step, calculate M and N for these steps and plot those in relation to the steps counter. Somehow this worked almost immediately with function generated strains:

But... why not existing software?

First off, input is easy! The functions for generating the graph, can be hidden. That way, input and graph will fit on one A4:

Next: the way the calculation has been poured into functions, makes it usable in MathCAD Solve blocks. In other words: we can use MathCAD to find the strains corresponding with random combinations of normal force and bending moments. Those strains can be translated to curvature, which can be used for calculating stiffness of the section:

Those two advantages however can also be achieved with other software I guess. The one I haven't seen though, which is really handy when designing a structure: combining multiple capacity curves, using the same functions. Neat right?

What's next? Calibrating the formulas maybe, to incorporate tension stiffening, optimize for SLS crack calculation, take into account buckling? It's all possible...