Concrete Column Interaction Diagrams

What is an Interaction Diagram?

In short, an Interaction Diagram is a much faster way of analyzing a concrete column for large eccentricities (aka large moments). An example of a Interaction Diagram has been included in Figure 1 (click the hyperlink to expand the image).

Figure 1: A Concrete Interaction Diagram (Click here for larger version)

Fantastic! Now how do I use it?

Don't fret, it's actually a lot simpler to read than it seems. I highly recommend you print out the large version of the Interaction Diagram that you can find here before continuing. It will clarify my explanation of how to use interaction diagrams greatly.

1) The constants:

Before you pick a interaction diagram you have to make sure you have the right one. That upper right hand corner of every diagram has a section that we'll call the constants for now. This section is how you differentiate one column from another. The following will usually be defined in the constants:

• f'_c = the strength of the concrete (e.g. 4 ksi)
• f_y = the yield strength of the steel in the column (e.g. 60 ksi)
• γ (gamma) = the ratio between the center-point of the steel reinforcement and the outside dimension of the column. For example, if you have #8 bars with a edge distance of 2" on a 20" x 20" concrete column then γ would be: [20" - 2*2" (cover) - 1" (dia. of bar)] / 20" = 15/20 = 0.75
• Lastly, it is important to differentiate between square tied columns and circular spiral columns in this section (you'll be able to tell the difference by the shape of the column in the upper right hand corner). In the example i've attached the column is a square tied column.

Note: If any of the variables fall between constants then you can conservatively take the lower of the two interaction diagrams for your numbers.

2) The horizontal and vertical axis:

As you can see the two equations below make up the horizontal and vertical axis' of the interaction diagrams.

where:

These two variables represent an Axial Force variable and a Moment variable (notice how one contains e/h and one does not). They are your starting point for figuring out where your preliminary point on the interaction curve is. See Section 5 below to see some hints for getting started.

Note: These variables are unitless so make certain that all of the units in both of the above equations cancel out.

3) The p_g and e/h lines:

Two additional pieces of information on these graphs are the p_g and e/h lines. These lines provide boundary conditions once you figure out your vertical and horizontal axis variables. As you can see, p_g represents the amount of steel in the column. The higher the moment you have, the more steel you will need. Steel will also greatly increase the axial capacity of a concrete column.

The e/h line is a shortcut to not take the moment acting on the column into account. It is not necessary to solve for both the e/h line and the horizontal axis (notice how they both have e/h, which is a variable that takes the moment acting on a concrete column into account. Since I always like solving for both axis' I usually ignore this step.

Note: An e/h of 1.0 represents a much larger moment than an e/h of 0.10

4) Why is ε on there?

In order to plot a point on the graph, you have to assume a phi (ϕ). The ε _t = 0.002 (compression controlled section) and e_t = 0.005 (tension controlled section) are on the diagram to be used as guides when assuming a value for your strength reduction factor. At this point you may have to revisit your horizontal and vertical axis variables and update your location on your interaction diagram (if you originally assumed compression controlled and it ends up being somewhere between the two).

5) Any other hints for getting started?

Of course!

• To get started assume a ϕ and check back later to make sure your assumptions were correct.
• A good place to be is to have p_g fall between 0.02 and 0.025
• If the point is outside the last curve (p_g = 0.08) then the section size is to small. The 8% steel content keeps the column from getting to congested.
• If your moment is inside the fist curve (p_g = 0.01) then your section size is to big.
• Transverse reinforcement can be found the same way it's found for columns with small eccentricity (see Concrete Tied Columns & Concrete Spiral Columns)

Where can i find them?

Most often these interaction diagrams are found in the Appendices of your Concrete books. Here are a few websites where I have found additional ones:

• http://www.struweb.com/interaction_order.html