Interaction diagram and failure modes

An interaction diagram is a curve (or a set of curves) that shows the combinations of axial load $P_u$ and bending moment $M_u$ that a given column section can carry at failure. Any pair $(M_u, P_u)$ lying inside the curve is safe; a point on or outside the curve means failure. It is drawn by moving the neutral axis from very deep (pure compression) to very shallow (pure bending) and finding $P_u$ and $M_u$ for each position from strain compatibility. SP-16 gives ready charts as non-dimensional plots of $P_u/(f_{ck}bD)$ against $M_u/(f_{ck}bD^2)$.

Features

   Pu
   ^
Puz*
   |   *          compression failure
   |      *       (small e, xu > xu,max)
   |         *
Pb |..........*   balanced point (Mb, Pb)
   |         *
   |      *       tension failure
   |    *         (large e, xu < xu,max)
 0 +---*---------------------> Mu
      M0 (pure bending)

• The top point $P_{uz}$ is pure axial compression ($M = 0$); the point on the $M_u$ axis is pure bending ($M_0$).
• The balanced point $(M_b, P_b)$ is where the concrete crushes ($\varepsilon_c = 0.0035$) and the tension steel yields at the same time. It separates the two failure zones, and the moment capacity is greatest there.
• Below $P_b$, the moment capacity increases as the axial load increases; above $P_b$, it decreases.
• A radial line from the origin has slope $1/e$, so it represents a constant eccentricity.
• Curves are plotted for different values of $p/f_{ck}$, for a given $d'/D$ and steel grade.
• A design point is safe if it lies inside the curve.

Modes of failure (eccentric compression)

1. Compression failure (small eccentricity, $P_u > P_b$, neutral axis deep, $x_u > x_{u,max}$): the concrete crushes first at the more compressed face, while the steel on the far side has not yielded. The failure is sudden (brittle) and lies on the upper part of the curve.
2. Balanced failure ($P_u = P_b$): the concrete crushes and the tension steel yields simultaneously.
3. Tension failure (large eccentricity, $P_u < P_b$, $x_u < x_{u,max}$): the tension steel yields first, followed by crushing of the concrete. The failure is gradual (ductile) and lies on the lower part of the curve.
4. Pure axial compression ($e = 0$) is a limiting case at the top of the curve. Since IS 456 requires every column to be designed for at least the minimum eccentricity $e_{min}$, the top of the design curve is cut off (for an axially loaded short column, $P_u = 0.4 f_{ck} A_c + 0.67 f_y A_{sc}$).