Column Interaction Diagram
Image by yours truly

Imagine this:

You woke up in the middle of the night bathed in perspiration. That one particular column, you just can't believe the size of which can withstand the forces it should carry. You thought you already checked all possible load combinations but there's one combination that only heaven knows why you forgot to encode in the software.

Dang old conscience hurts now. You decided to check it right then and there even if the penalty is a night of lost sleep. You perfectly remember anyway the axial loads and the magnified moments on both axes.

Just one problem.

Your laptop just bogged down and you have no software and no printed material whatsoever to access to an interaction diagram. Panic sets in. How in the world are you going to check that column so that you can be assured that it's going to be fine?

Ok, the above scenario may be overly dramatic and you may not be buying it so let us put it this way: on your way to structural engineering sensei-dom, we want to learn the basis of everything structural and leave the programming part to the software. So a new world order may be on the way but amid the rubbles after the big nuclear bang, you still know perfectly from memory how the dang thing works (you will need it when you start to build the new world.)

Ok so how does this work?

The point coordinates in the column-interaction diagram curve which are the corresponding axial-to-bending moment capacity envelope are derived based on different strain conditions listed below and their corresponding stress diagrams.

The said conditions are the following. It is best if you refer to this spreadsheet with calculations and detailed explanation in order to fully grasp my points.

Pt#1: Pure axial compression capacity. Zero bending and the axial capacity is based from equation 10.3.6.2 (10-2). Reduction factor φ = 0.65

Point 1
Point 1. Snapshot form ACI 318M-11

Pt#2: Maximum axial compression and its corresponding bending moment capacity. In the example, the neutral axis is outside the column cross-section which is an indication that the entire cross-section is under compression. Reduction factor φ = 0.65

Point 2 Condition. Drawn by yours truly.

Pt#3: Zero strain at concrete edge. Under the combined axial and bending, the cross section is now on the verge of take off. Albeit, in this condition all the steel are still in compressive strain. Reduction factor φ = 0.65

Point 3
Point 3 Condition

Pt#4: Zero strain at extreme steel reinforcement. A part of the concrete is now in tension and the extreme steel reinforcement is already at zero, meaning a part of the reinforcement is now on the verge of taking the tension. Reduction factor φ = 0.65

Point 4
Point 4 Condition

Pt#5: 0.002 strain at extreme steel reinforcement. From zero, the tensile strain now rose to 0.002. Now the significance of this 0.002 strain is that it marks the end of the compression controlled zone. Meaning, beyond the tensile strain of 0.002 (and not greater than 0.005) there is now a transition from the load reduction factor of φ = 0.65 as per Fig. R9.3.2 below. However, at exactly 0.002 strain, the reduction factor is still φ = 0.65.

Point 5
Point 5 Condition
Point 5b
How to interpolate the reduction factor. Snapshot from ACI 318M-11

Pt#6: This is the balanced condition where the concrete and the steel simultaneously reach their maximum compressive strain and yield strains respectively. We are now on the transition zone such that we now have to make use of the interpolation from the figure above. At the strain of 0.0023, the reduction factor is now computed as φ = 0.675 (you can try and check the interpolated value).

Point 6
Point 6 Condition

Pt#7: We've now reached the tensile strain of 0.005. This now marks the end of the transition zone and into the tension zone where the φ = 0.9 for strains equal to or greater than 0.005

Point 7
Point 7 Condition

Pt#8: Pure bending. The section now is under pure bending (net axial load on the section is equal to zero). Load reduction factor is φ = 0.9

Point 8
Point 8 Condition

Pt#9: Pure axial tension. The concrete is now in full reliance to the steel to resist tension. Load reduction factor is φ = 0.9

Point 9
Point 9 Condition

Verification

I verified my hand calculations using Prokon and ETABS and a spreadsheet developed for column design. For ETABS, I extracted the points of the capacity curve and overlayed it in the points I derived from hand calculations.

Comparison

You can do your own programming or extend it in Excel. But with the highlighted 9 points, you now have a basically complete interaction diagram that you can use. All you have to do now is plot the applied loads (I wish it is within the capacity curve!) Also, don't forget to magnify the moments where required.

Now in case you missed my spreadsheet which I mentioned above, you can download it here.