Singly reinforced beam - strain development

[thoughtexplorer]

[URL unfurl="true"]https://res.cloudinary.com/engineering-com/image/upload/v1712223175/tips/New_Drawing_4_ouv9sq.pdf[/url]

Check my understanding.

when a RC beam is subjected to bending, both the concrete strain (Ɛc) and reinforcement strain (Ɛs) started out the same, say 0.0005.
As moment increase, both Ɛc and Ɛs will also increase, say 0.001.

But eventually, reinforcement yields at Ɛs = 0.00217 (yield value is based on Eurocode).
At this stage, concrete strain should also be Ɛc = 0.00217 due to equilibrium.
The depth of neutral axis (x) here will be 0.5d (where d is effective depth).

Now from here on, if we were to keep increasing the applied moment, Ɛs will still be 0.00217 where it yields.
But concrete could still be further strained up to its ultimate value Ɛc = 0.0035.
At this stage, the depth of neutral axis (x) has shifted down to 0.617d.

This window between 0.5d and 0.617d where concrete strain (Ɛc) varies from 0.00217 to 0.0035 (a theoritical value) is not a risk engineers want to take as the concrete could crush anytime. Hence, codes limit neutral axis depth (x) to 0.45d (Eurocode).

That's why the allowable concrete compressive strength for flexure design in Eurocode is based on a concrete stress block depth derived from x = 0.45d.
If the compressive force on the concrete induced by the applied moment > the allowable concrete compressive strength, then we will either need to do doubly-reinforced or increase the depth.

In summary, if an applied moment causes the beam to behave such that depth of neutral axis > 0.45d, increase beam depth or go with doubly-reinforced design.
Let me know if my understanding is correct. If it is, then I have actually came across multiple singly-reinforced beam design examples where x > 0.45d but < 0.617d when I back calculated the x. This is confusing because it contradicts with the whole idea of limiting x = 0.45d.

Let me know if you have any thoughts. I'm a university student majoring in structural engineering.

 

[Retrograde]

You should post this in the student engineering section of the forum.

 

[cam_b]

I agree this should be in the student section.

I would say that your understanding is not quite correct. My understanding is as follows, the force is in equilibrium, not the strain in the steel and concrete. As the tensile steel and extreme compressive concrete face are generally not the same distance from the neutral axis. The neutral axis is not simply taken as 0.5d especially once it has cracked. Its location needs to be worked out. In a cracked section, if the tensile steel is in the bottom then it is generally much higher up than than 0.5d from the bottom.

Secondly, i would group the behaviour of an R.C. section into three distinct phases, uncracked, cracked elastic then cracked plastic. You can use the modular ratio to calculate an equivalent section and see how it would behave when its uncracked. When its uncracked the steel has fairly low tensile forces, as its generally low area compared to the concrete mean that it will not attract particularly large forces. However, once the concrete exceeds its flexural tensile limit, it cracks and all the tensile force is now taken by the steel. Hence minimum steel requirements. To ensure sections do not fail suddenly when the concrete reaches its cracking moment. Once cracked, if the bending moments increase further, the tensile steel will eventually yield and behave in a plastic manner, so the strain in the steel may increase, but the force in the bars will not. The extreme compressive concrete will also behave in a plastic manner. Hence many codes adopting a concrete compressive block, this block is approximating the stress distribution in the concrete once most of the compressive zone is behaving plastically. When going from cracked elastic to cracked plastic, you may notice that the forces in the materials have not increased, but the moments have. This additional moment capacity is because the neutral axis moves up in the section (assuming tension in the bottom) therefore the lever arm between the resultant tension and compression forces has increased.

Limits on neutral axis depth are commonly used as over-reinforcing sections can lead to non-ductile failure as the concrete fails before the steel yields.

 

[rapt]

You forgot that concrete cracks!