Stress & strain for tension & compression members

[StrEng007]

There is a theoretical concept I'm trying to straighten out, which came up when I was looking at the stress vs. strain curves for concrete cylinder compression tests. Before I get to that question, I'd like to briefly touch upon the diagrams we typically see for tensile tests.

First off, I understand the reasoning behind engineers using an "Engineering" stress vs. strain approach to the diagrams as opposed to the true stress vs. strain. It does greatly simply the analysis when we consider the original areas and original lengths in our calculations. However, the part of these typical "engineering" curves that are counter intuitive to me... are the locations where necking and fracture is occurring. For instance, consider ductile steel materials. Many texts will reference, with the idea of engineering s/s in mind, that the breaking strength of a material is less than the ultimate strength. However, when you consider that the material is going to decrease in actual cross sectional area, the stress at the point fracture will indeed be much higher as indicated in a "true" stress vs. strain diagram.

The reasoning I've come to understand is that, the engineering curves always represent the behavior with the original area and length in mind. Therefore, if the true area is decreasing, while the length is increasing, while the stress is increasing... we compensate for it by always considering the original geometry. Doing so gives a graphical image that the stress decreases prior to fracture, when it actually doesn't. *Unless the test unloads the specimen in relation to constant strain, then that would make sense*

I've read many resources online but haven't found one yet that can "simply state" (simply and elegantly) why the engineering stress strain curve drops off before fracture. I suppose for the tensile applications, it doesn't even matter because we use values back at the ultimate strength point of the curve. So there is no use getting all bothered about it.... But here's the situation that came up for me:

If you look at a compression stress vs. strain diagram for concrete, you'll notice the same drop off at a strain roughly equal to 0.003. With that being said, the inverse of the tension test is happening here... the concrete is shorting and by Poisson's ratio the area is increasing. Yet, we see the same drop off? If the same idea as the tensile test is applied, wouldn't adjusting the areas spike the stress? I'm not sure if these tests, tension or compression, adjust the test force to keep the strain constant through the process, meaning they actually decrease the force as the materials are entering their breaking/rupture states. This would explain why we see that drop off for concrete.

The reason why I bring this up is because we use a concrete strain at 0.003, and not 0.002 where the ultimate strength of concrete is typically derived from. In order to consider the 0.003 strain, I want to clearly understand what the stress is doing. Why not use 0.002 for the strain of concrete as it is more closely related to the ultimate strength?

 

[rapt]

We use .003 (or other values depending on the code) at the extreme compression face, reducing to 0 at the neutral axis. This is a simplified guess at the concrete strain at the peak of a moment curvature diagram for the cross-section. You can always generate the moment curvature diagram and determine the peak capacity if you want and the concrete strain at that point on the diagram if you want to, but you will never get your designs finished!.

For steel, we do not use the peak capacity, we use the yield capacity which is approximately the point at which the steel stress/strain curve goes plastic.

 

[StrEng007]

Thanks for the input.

The 0.003 is not a "simplified guess" at the concrete strain at the peak of moment curvature diagram. The process you've mentioned is a sequence in which the limiting point of the moment-curvature diagram is reached when the concrete top strain is set equal to εu which is referenced by ACI 10.2.3 to be equal to 0.003. My question is not in regards to this diagram as we already know that ACI requires use of 0.003.

My question is in regards to WHY we're using 0.003 when graphically, and as referenced by ACI R10.2.6, that the ultimate compression strength of the concrete is seen at stain values of 0.0015 to 0.002.

After giving this some though, while it looks like the stress-strain diagram "drops off" to 0.003, the stress in the compression cylinder is higher than at f'c. I had thought Poisson's ratio would have provided an inverse curve in the "engineering stress/strain" diagram due to enlarging the area as referenced above. However, similar to the tensile test of steel, the concrete cross sectional area prior to failure is becoming smaller due to spalling and cracking. What made this evident to me is seeing a comparison of tied column to spiral column stress strain diagrams... the spiral diagram does not drop off. Reason being, the spiral tie is excellent in maintaining the cross sectional area while the tied column will lend to a severe decrease in cross-sectional area and rapid failure.

Seeing a true stress vs. strain diagram will shed some light on why the engineering stress strain diagrams look the way they do.

 

[rapt]

My reply IS the reason why .003 is used!!!!

 

[Hokie93]

The "maximum usable strain" (ACI 318-11 10.2.3) equal to 0.003 has been selected based on failure of the concrete, not the strain associated with peak stress. This is pretty well explained in R10.2.6. The concrete textbook by Phil Ferguson (an oldie but goodie) has a good discussion.

 

[rapt]

StrEng007

To generate a moment curvature diagram for a cross-section, using a curvilinear stress/strain diagram for concrete (not rectangular stress block) you start with a small strain value and determine the section capacity. Then increment the strain value determining the section capacity at each increment. Plot these on a diagram. You will notice that the curve will reach a peak moment capacity and any increases in strain value after the peak will result in lower capacity. The strain where the peak moment capacity is reached is used as the ultimate strain.

ACI has nominated a value of .003 for this strain at peak capacity to save designers having to go through the process on every design. Eurocode has nominated .0035 for lower strength concretes dropping to .0028 at about 80MPa concrete strength, recognising that the stress/strain relationship for higher strength concretes is closer to a straight line and has a far shorter zone past the peak compression point. In fact it assumes that at about 100MPa there is no drop off after the peak.