Design steps for long column, unbraced frame

A column is long (slender) when $l_{ex}/D$ or $l_{ey}/b$ exceeds 12. In an unbraced frame (sway frame) the lateral stability depends on the columns themselves, so the effective length is larger than the unsupported length and the additional moments due to slenderness must be added (IS 456 cl. 25.1.2, 39.7).

Design steps

1. Data: size $b \times D$, factored loads $P_u$, end moments $M_{ux1}, M_{ux2}, M_{uy1}, M_{uy2}$, unsupported length $L$, grades of concrete and steel.
2. Effective length: for a sway frame, find $l_{ex}$ and $l_{ey}$ from the charts of IS 456 Annex E (Fig. 27 / Fig. 28) using $\beta_1, \beta_2$ (ratio of column stiffness to beam stiffness at both ends), or from Table 28 when end conditions are known.
3. Slenderness ratio: $l_{ex}/D$ and $l_{ey}/b$. If either is above 12, the column is slender about that axis; if above 60, redesign (cl. 25.3.1: $l/b \le 60$).
4. Minimum eccentricity: $e_{min} = \dfrac{L}{500} + \dfrac{D}{30} \ge 20$ mm for both axes.
5. Assume a trial steel percentage (say 1% to 3%) and bar arrangement; $d'/D$ for cover.
6. Additional moments due to slenderness for the slender axis: $$M_{ax} = \frac{P_u D}{2000}\left(\frac{l_{ex}}{D}\right)^2, \qquad M_{ay} = \frac{P_u b}{2000}\left(\frac{l_{ey}}{b}\right)^2$$ and reduce by $k = \dfrac{P_{uz} - P_u}{P_{uz} - P_b} \le 1$, with $P_{uz} = 0.45f_{ck}A_c + 0.75f_yA_{sc}$.
7. Total design moments: in an unbraced column the additional moment is added to the larger end moment, and the end moment is not reduced (for a sway frame there is no reduction for single/double curvature): $M_{ux} = M_{ux,2} + M_{ax}$, $M_{uy} = M_{uy,2} + M_{ay}$, not less than $P_u e_{min}$.
8. Check the section: find $M_{ux1}, M_{uy1}$ (uniaxial capacities at $P_u$) from SP-16 charts, and check the biaxial interaction $$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} \le 1$$ with $\alpha_n$ from 1.0 ($P_u/P_{uz} \le 0.2$) to 2.0 ($\ge 0.8$) (cl. 39.6).
9. Revise: if not satisfied, increase steel (limit 0.8% to 4%, maximum 6%) or the size, and repeat from step 6.
10. Lateral ties: diameter $\ge \max(\phi/4, 6)$ mm, pitch $\le \min(b, 16\phi, 300)$ mm; in a seismic zone use ductile detailing (IS 13920).
11. Draw the section and elevation, showing bars, ties and laps.