Significance of under and over-reinforced sections

Under-reinforced section

• Steel area is less than the balanced value, so $x_u<x_{u,max}$ and steel yields before concrete crushes.
• Failure is ductile: wide cracks and large deflection warn the occupants, and the beam can redistribute moment.
• Economical, since the high strength of steel is fully used. Moment $M_u=0.87f_yA_{st}d(1-A_{st}f_y/bdf_{ck})$.

Over-reinforced section

• Steel area is more than balanced, so $x_u>x_{u,max}$; concrete reaches 0.0035 while steel is still elastic.
• Failure is brittle and sudden, with few cracks and no warning. Steel strength is wasted.

 under-reinforced        over-reinforced
 strain: es > ey         strain: es < ey
 |\ 0.0035               |\ 0.0035
 | \ xu < xumax          | \  xu > xumax
 +--  NA                 +----  NA
 |   \ es (large)        |  \ es (small)
 Ductile                 Brittle

Significance in design

1. The code limits $x_u\le x_{u,max}$, so every designed section is under-reinforced or at most balanced.
2. If $M_u>M_{u,lim}$, increase depth/width or provide compression steel (doubly reinforced) instead of more tension steel.
3. Maximum steel is limited to 4% of the gross area; minimum steel $0.85bd/f_y$ prevents sudden failure on cracking.
4. Ductile behaviour is essential in earthquake-resistant design.