Under, over and balanced sections

In limit state design, the type of beam section is decided by the depth of neutral axis $x_u$ compared with the limiting depth $x_{u,max}$ (IS 456, cl. 38.1): $0.53d$ (Fe250), $0.48d$ (Fe415), $0.46d$ (Fe500).

Under-reinforced section ($x_u<x_{u,max}$)

• Steel area is less than that for the balanced section; steel yields before concrete reaches its ultimate strain 0.0035.
• Failure mode: ductile. Steel yields, cracks widen and deflection becomes large, giving ample warning before concrete crushes.
• Moment: $M_u=0.87f_yA_{st}d\left(1-\dfrac{A_{st}f_y}{bdf_{ck}}\right)$.
• Preferred in design.

Balanced section ($x_u=x_{u,max}$)

• Steel yields and concrete reaches 0.0035 at the same time.
• Moment: $M_{u,lim}=0.36f_{ck}b\,x_{u,max}(d-0.42x_{u,max})$, i.e. $0.138f_{ck}bd^2$ for Fe415.

Over-reinforced section ($x_u>x_{u,max}$)

• Excess steel; concrete crushes (strain 0.0035) before steel yields.
• Failure mode: brittle and sudden with no warning. Not allowed in design; capacity is limited to $M_{u,lim}$ and compression steel is added if more moment is needed.

  Section     Strain             Stress block
  +----+                         0.446fck
  |    |   0.0035               ||====||
  |    |   |\                    ||    ||
  |  x |   | \ xu        xu      ||    ||
  |    |   |  \                  ||    ||
  +----+  -+---  NA            C
  | As |   |  /
  +----+   | / eps_s >= ey (under), = ey (balanced), < ey (over)