# Why does rebar in reinforced pre-stressed conrete reduce in deflection if compressive stress is applied to member?

According to What is Prestressed Concrete? - Practical Engineering, one way to reduce deflection in the steel rebar is to pre-stress the rebar by applying compressive stress in the member before service which is done by tensioning the steel rebar.

My question is: Why does applying compressive stress to member decrease deflection?

especially with relation to the formula: $$y=\frac{FL^3}{48EI}$$

where

• $$y$$ is deflection
• $$F$$ is force applied to member at $$L/2$$
• $$L$$ is length of beam
• $$E$$ is elastic modulus
• $$I$$ is second moment of area

Prestress creates an upward deflection on the beam due to the combination of elastic strain, creep and shrinkage in the unloaded beam.

This negative deflection will bend back to a straight stance or at least minimize much of the deflection of the beam after the loading.

The equations of the deflection of a prestressed concrete beam depend on the curvature or geometry of the prestressed bars and crack depth.

Because of this each case has to be analyzed depending on its configuration.

There are design aids that have many common rebar curves and bends.

Concrete is strong in compression but weak in tension. In tension, it will crack at a much lower stress than the allowable compression stress.

A reinforced concrete beam without any prestressing will generally have compression in concrete and tension in the rebars. If there are any significant tension stresses in the concrete, the concrete will crack and the rebars will take over, transferring the tension instead of the concrete. Since the part of the concrete which is cracked can no longer transfer any tension stresses, it will not contribute to the second moment of area.

If the concrete beam had a large compression force from prestressing, there would be no tension stresses and therefore also no cracking. That means the entire concrete cross section contributes to the second moment of area.

So in terms of the deflection formula, you quoted in the question, the answer is that the second moment of area is different, because we avoid the cracking.