We have seen the stress-strain curve for concrete, and we have seen the stress-strain curve for steel. But do we ever notice, that these two curves are actually on the complete opposite quardrant of the chart – the concrete one is for compression and the steel one is for tension. There is never a stress-strain curve for concrete in tension, nor one for steel in compression. Further more, there are so many different versions of stress-strain curve for concrete, just to perfectly confuse us. This blog post aims to have a step-by-step anatomy of the stress-strain curve of concrete and clarify every detail and assumption behind the various concepts flying around in Eurocode. There are a lot of things we can talk about so I plan to go through everything in two blogs.
Stress-strain curves
We surely have seen these two diagrams for idealised concrete stress-strain curve. They are both elasto-plastic curves – once the material reaches its maximum capacity, the stress remains constant whilst the strain infinitely increases. We’ll come back to this in the next blog post, to talk about their differences.
We may have also seen this curve given by Eurocode, which is much closer to the true behavior of concrete:
Let me re-plot this diagram for clarity:
This is a concrete stress-strain curve commonly seen. Hardly ever noticed, there are three very important assumptions for this diagram:
The concrete is subject to an uniaxial static point compressive load.
The concrete is unreinforced and unconfined.Confinement of concrete results in a modification of the effective stress-strain relationship: higher strength and higher critical strains are achieved.
The loading is short term. Long term loading activates time-dependent behavior of concrete such as creep.
If any of these assumptions are invalidated, the stress-strain curve will change.
This diagram tells us that the concrete behavior starts to be non-linear even at small stress. The linear relationship is only idealised. 𝜀𝑐u is the ultimate compressive strain.
We can notice the strain softening behaviour of concrete once the curve reaches its peak at 𝜀𝑐1. 𝜀𝑐1 is the compressive strain at maximum stress. 𝜀𝑐1 gets bigger as the concrete grade increases. In other words, higher grade concrete will be able to undergo larger strain before it starts to soften. Strain softening happens to densely compacted granular materials – high stress breaks apart the already well-organised structure of the material making it loose and hence softer. Loosely compacted soil, on the other hand, is a strain-hardening material – high stress forces its granular structure to condense into a better organised structure.
So going over the ultimate strength (fcm) of concrete is a very dangerous thing. To prevent that from happening, we must limit the strength of concrete allowed in our design to a lower value. Here brings out the concept of ‘characteristic value’ – see later in this blog. In the design, we start with characteristic values and factor it down to a safe value, so as to prevent concrete from reaching its ultimate strength.
Concrete grade – what does it really mean?
The ultimate strength (fcm) of concrete depends on the grade of concrete. The concrete grade usually takes the form of C32/40, C45/55 etc, which we see on drawings and documents. But what does it really mean? What are the assumptions behind? What wriggle room do we have as designers?
The full name for the so-called ‘concrete grade’ is: Characteristic cylindrical/cubic compressive strength of concrete at 28 days.
The cube strength is based on 150mm by 150mm cube tests, whereas the cylindrical strength is based on 300mm tall by 150mm diameter cylinders. It is not difficult to anticipate that the concrete appears to be stronger for cube tests. The cubic strength is about 1.25 times that of the cylindrical strength.
Also, strength of concrete increases over time. What is used in designs are strengths normally at 28 days. Concrete continues to gain strength beyond 28 days, although at a significantly lower rate. The rate of increase normally can be considered negligible once the concrete passes the 90 days old mark.
Another important concept is the ‘characteristic value’. In Eurocode, for material properties, the characteristic value means the value set at a level such that it is achieved in real life 95% of the time statistically. Note that the non-linear stress-strain curve of EC2 uses the ‘mean ultimate stress’, which is higher than the characteristic one because it is set at a level such that it is achieved 50% of the time statistically. The difference between the mean value and the characteristic value can be understood as a kind of standard deviation. This ‘standard deviation’ remains constant for all grades of concrete (8MPa).
Adding characteristic value and the design value to the diagram, it becomes:
Concrete in tension
When unreinforced concrete is subject to tension, the failure is brittle and there is no softening behaviour. It just snaps, then the stress falls off the cliff. This is exactly why concrete needs reinforcement! For the same reason, tensile strength of concrete is not permitted to be utilised for Ultimate Limit State. Nevertheless, from a stress-strain behaviour perspective, there is still residual stiffness of concrete where reinforcement is present, even after cracking, and this is called 'tension stiffening'.
Concrete behaviour in tension is a large subject in its own right. Refer to this post for details:
If we add tension into the diagram, it becomes like this:
More interesting details to be continued in the next post