To understand the importance and necessity of a doubly reinforced beam, first we should look at a singly reinforced beam.
A beam when loaded can possibly suffer through two different type of failure:
1. Ductile Failure
2. Brittle Failure
Image showing significant cracking in tension and still the compression block is not showing any kind of failure. This can lead to ductile failure which is our priority
Now, in case of a brittle failure when the beam gets overstressed, then till the point of over-stressing you will not see any major cracks in beam, but as the point is passed and suddenly (BAM..!!!) the beam fails. This is no good right? So we always want the beam to fail in a ductile fashion as it will give you more time to see that the beam is over-stressed and it is time that you do some repairs or evacuate the house.
Now how do we make sure that the failure should remain ductile. This can happen only when the strength of steel is less than the compressive strength of concrete. This means that when ultimate loads are reached, as the total strength of steel is pretty less than strength of concrete, steel will try to strain more than what concrete will experience strain in compression. Now we know that steel is ductile and can handle higher strains without any kind of failure. So this is how we get a ductile response of beam. Concrete is all under control while steel is straining itself.
But suppose if I tell you that you cannot use a section greater than say 24" x 24" and you have very high moments to deal with. This high moment will tend to increase the tension steel demand and because of this a bigger concrete block will be required in compression which will push the neutral axis further down. Now this is important part. You may ask yourself that so what if the neutral axis go further down, I have a whole concrete beam which can take compression. But wait..!!
As the neutral axis starts shifting down, the strains in extreme fiber starts to increase and this will be our concern in case of compression fiber. Concrete is brittle, so at first it will not show that much impact, but this high value of strain will cause concrete to crush itself which will lead to ultimate failure and this will be instantaneous. So there you go, you have your answer to why we cannot put more steel than a certain amount.
Image showing brittle failure of beam as the compression stress block has failed and there is no significant cracking in tension
Now what decides this limit of steel? Well, it is all experimental based approach and to an extent you can prove it mathematically too balancing tension and compressive forces, take the depth of compression block, then draw the strains and see if the strain in concrete is more than the allowable strain. But if you are designing it, then some codes specify the maximum reinforcement ratio in a beam while some codes specify the maximum possible neutral axis depth that a concrete beam can achieve, if you are under this, then you will see a ductile failure in a beam and you are safe.
Talking about double reinforced beams:
Now as I mentioned before that what if you are limited to a 24" x 24" section and you have pretty heavy moment and you are exceeding the limits mentioned in the codes, so the only option is to provide compression reinforcement and add some tension steel. Now the extent to which you are adding the compression reinforcement will tell you whether the beam is brittle or ductile.
Suppose the limit on tension reinforcement for a singly reinforced beam is X, and to resist this moment you have to add additional 0.5X of tension reinforcement. You also decide to add 0.8X of compression reinforcement. So now you have 1.5X of tension reinforcement and 0.8X of compression reinforcement along with the compression stress block. So now what will happen is this 0.8X of compression reinforcement will tend to balance the effect of 0.5X of tension reinforcement. I have added 0.8X of compression reinforcement as the beam does not experience a very high compressive strain as it experiences the tension strain. So the compression bars will be under a lower stress than its yield point. Thus to balance a fully stressed 0.5X of tension reinforcement I will need a higher amount of compression reinforcement. Now, the concrete block is responsible to resist the X amount of tension reinforcement which is the limit for ductile behavior. So in this case the doubly reinforced beam will act as a ductile beam and so you can say that it is an under-reinforced section. All good..!!
But suppose you decide to add only 0.4X of compression reinforcement. This will counter a max of 0.4X of tension reinforcement in worst case scenario. So now, a plain concrete stress block is responsible to resist 1.1X of tension reinforcement and this is above the limit which will result in brittle failure. So in this case, the beam, even though it is doubly reinforced, it will experience a brittle failure which means an over-reinforced section.
To conclude our discussion, not all doubly reinforced sections are under-reinforced. It depends on the trade-off between the amount of compression steel provided as compared to tension steel.
I hope you enjoyed our little discussion.