# Reinforced Concrete Columns - Interaction Diagram Question

I'm confused about some concepts concerning interaction diagrams for reinforced concrete columns. Hopefully somebody will be able to answer these:

1. Point with zero tensile strain, but before uniform compression (no moment)

In the calculation for this point, I see that the distance $c$ (distance to neutral axis from extreme fiber) is assumed to be the height of the cross section. I understand that the strains are all compressive which means that no tensile zone will exist below the neutral axis, but why wouldn't it be assumed that $c = d$ (distance from extreme fiber to last layer of steel). Since the analysis of columns with axial + moment is essentially an extension of beam analysis, why would the concept of effective depth not be used? In other words, I think the strain distribution for this point should act from the extreme fiber to the last layer of steel.

1. Stresses in the steel layers are computed based on

a) An assumed value for $c$ (for the zero tensile strain point with moment)

or

b) A calculated value for $c$ (for the balanced point and below)

Based on this value of $c$ when the stress in the steel layers is computed it can come out to be less than, equal to, or greater than the yield stress for the corresponding reinforcement. I don't understand how this can happen. In beam design a strain distribution is assumed (under, over, or balanced), and forces are computed based on the stress in the steel. The assumption you make determines whether you initially know the stress in the steel or if it has to be computed. Either way since the steel is assumed to be elastic-perfectly plastic, the stress in the steel is never going to be greater than the yield stress. Based on this concept, how can the stress in the steel come out to be greater than the yield stress when forming the interaction diagram. From what I can tell if the stress in the steel comes out to be greater than the yield stress, the stress in the steel is set equal to the yield stress for the purpose of computing forces. Can someone explain this iterative process for computing the steel stresses?

• Can you clarify what you mean by "Point with zero tensile strain but before uniform compression (no moment)". Which point on the interaction curve are you referring to exactly? The point of pure axial compression and no moment? – CableStay Oct 15 '16 at 21:51

To start off, I thought I'd just throw up a conceptual sketch of an interaction diagram as a point of reference.

Re: Question #1 --- My best guess is that you're referring to a location in the compression controlled region where the strain profile is as in Case A, below.

As you state in your question: By definition, $c$ is the distance from the extreme fiber (at the assumed compressive strain of 0.003) to the neutral axis. To produce the interaction diagram, we assume values of $c$ and then use equilibrium equations to calculate $P$ and $M$.

Therefore, there is a point on the interaction curve where $c$ is the distance from the extreme fiber to the lowest layer of steel (Case B, below). Now the concrete below the steel is in tension, the concrete is assumed to be cracked, and the strength contribution of that concrete will be ignored.

I suppose theoretically, there is a small zone on the interaction curve where the concrete below the steel layer is in such a small amount of tension that the concrete could be uncracked. But the convention in practice is to assume concrete has no tension capacity.

Re: Question #2 --- You are correct that in practice we assume elastic-perfectly-plastic material properties for the steel (as shown below) and therefore, while the strain may continue to increase, the stress in the steel cannot exceed the yield stress.

Remember that in creating the interaction diagram, we're starting with a strain profile. The tension strain in the steel simply is whatever it is for a given neutral axis location. Then you're probably using Young's Modulus ($E$) to calculate the corresponding stress in the steel. By doing this, you're assuming a perfectly elastic material. Since we know we're actually elastoplastic, whenever $E\epsilon$ is greater than the yield stress, we can say that the steel is actually just at the yield stress.