Design of compression members: Columns

Check of the given section 350 × 350 mm

Even with the largest practical steel (for example 14-25 mm bars, $p = 5.6\%$), the interaction ratio is 1.77 > 1 for $P_u = 1440$ kN with the given moments. The given section is therefore too small and the column must be enlarged.

Data and assumptions

• The section given in the question (350 × 350 mm) is checked first and found inadequate (see the first check), so a revised section 450 × 450 mm is designed.
• Column 450 mm (width $b$) × 450 mm (depth $D$); $P_u = 1440$ kN; $f_{ck} = 20$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 195 kN·m; about minor axis (y-y) $M_{uy}$ = 180 kN·m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3600/500 + 450/30 = 22.2$ mm (min. 20 mm); adopt 22.2 mm; $M_{ux,min} = P_u e = 32.0$ kN·m
• About y-y: $e_y = 3600/500 + 450/30 = 22.2$ mm (min. 20 mm); adopt 22.2 mm; $M_{uy,min} = 32.0$ kN·m

Design moments

• Design moments: $M_{ux} = 195.0$ kN·m, $M_{uy} = 180.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1440000}{20 \times 450 \times 450} = 0.356$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 196610 + 0.75 \times 500 \times 5890 = 3978$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1440}{3978} = 0.362$, so $\alpha_n = 1.27$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{195.0}{330.6}\right)^{1.27} + \left(\frac{180.0}{330.6}\right)^{1.27} = 0.974 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-25 mm bars ($A_{sc} = 5890$ mm², $p = 2.91\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(450, 400, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Ductile detailing of transverse steel (IS 13920)

• Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(450, 600, 450) = 600$ mm → provide 600 mm from each joint face.
• Spacing of hoops within $l_o$: $\le \min(b/4, 100) = 100$ mm and not less than 75 mm → 100 mm c/c.
• Area of hoop leg: $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, with $h = 370$ mm (longer dimension of hoop, 40 mm cover), $A_k = 136900$ mm² (core to outside of hoop), $A_g = 202500$ mm². $A_{sh} = 0.18 \times 100 \times 370 \times \dfrac{20}{500} \times \left(\dfrac{202500}{136900} - 1\right) = 127.7$ mm²
• 3 legs of 8 mm give 151 mm² $>$ 127.7 mm², so a 8 mm hoop with cross-ties (3 legs in the direction considered) is adequate. Hoops have 135° hooks with 6φ (≥ 65 mm) extension.
• Outside $l_o$, ties at spacing $\le$ 225 mm, i.e. 220 mm c/c is provided.

Detailing

CROSS-SECTION 450 x 450
    +-------------------------+
    |                         |
    |  o      o     o      o  |
    |                         |
    |                         |
    |  o                   o  |
    |                         |
    |  o                   o  |
    |                         |
    |                         |
    |  o      o     o      o  |
    |                         |
    +-------------------------+
   o = 25 mm bars, ties 8 mm @ 300 c/c

Answer: 450 × 450 mm column with 12-25 mm longitudinal bars ($p = 2.91\%$) and 8 mm ties @ 300 mm c/c; hoops 8 mm @ 100 mm c/c over 600 mm at each end.

Check of the given section 300 × 400 mm

Even with the largest practical steel (for example 14-25 mm bars, $p = 5.7\%$), the interaction ratio is 1.34 > 1 for $P_u = 1400$ kN with the given moments. The given section is therefore too small and the column must be enlarged.

Data and assumptions

• The section given in the question (300 × 400 mm) is checked first and found inadequate (see the first check), so a revised section 400 × 500 mm is designed.
• Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1400$ kN; $f_{ck} = 20$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 200 kN·m; about minor axis (y-y) $M_{uy}$ = 110 kN·m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3000/500 + 500/30 = 22.7$ mm (min. 20 mm); adopt 22.7 mm; $M_{ux,min} = P_u e = 31.7$ kN·m
• About y-y: $e_y = 3000/500 + 400/30 = 19.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 28.0$ kN·m

Design moments

• Design moments: $M_{ux} = 200.0$ kN·m, $M_{uy} = 110.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1400000}{20 \times 400 \times 500} = 0.350$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 195602 + 0.75 \times 500 \times 4398 = 3410$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1400}{3410} = 0.411$, so $\alpha_n = 1.35$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{200.0}{292.9}\right)^{1.35} + \left(\frac{110.0}{234.0}\right)^{1.35} = 0.958 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 14-20 mm bars ($A_{sc} = 4398$ mm², $p = 2.20\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(20/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 320, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Ductile detailing of transverse steel (IS 13920)

• Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(500, 500, 450) = 500$ mm → provide 500 mm from each joint face.
• Spacing of hoops within $l_o$: $\le \min(b/4, 100) = 100$ mm and not less than 75 mm → 100 mm c/c.
• Area of hoop leg: $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, with $h = 420$ mm (longer dimension of hoop, 40 mm cover), $A_k = 134400$ mm² (core to outside of hoop), $A_g = 200000$ mm². $A_{sh} = 0.18 \times 100 \times 420 \times \dfrac{20}{500} \times \left(\dfrac{200000}{134400} - 1\right) = 147.6$ mm²
• 3 legs of 8 mm give 151 mm² $>$ 147.6 mm², so a 8 mm hoop with cross-ties (3 legs in the direction considered) is adequate. Hoops have 135° hooks with 6φ (≥ 65 mm) extension.
• Outside $l_o$, ties at spacing $\le$ 200 mm, i.e. 200 mm c/c is provided.

Detailing

CROSS-SECTION 400 x 500
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 20 mm bars, ties 8 mm @ 300 c/c

Answer: 400 × 500 mm column with 14-20 mm longitudinal bars ($p = 2.20\%$) and 8 mm ties @ 300 mm c/c; hoops 8 mm @ 100 mm c/c over 500 mm at each end.

Data and assumptions

• Column 350 mm (width $b$) × 500 mm (depth $D$); $P_u = 1500$ kN; $f_{ck} = 30$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 130 kN·m; about minor axis (y-y) $M_{uy}$ = 80 kN·m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3300/500 + 500/30 = 23.3$ mm (min. 20 mm); adopt 23.3 mm; $M_{ux,min} = P_u e = 34.9$ kN·m
• About y-y: $e_y = 3300/500 + 350/30 = 18.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 30.0$ kN·m

Design moments

• Design moments: $M_{ux} = 130.0$ kN·m, $M_{uy} = 80.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1500000}{30 \times 350 \times 500} = 0.286$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 30 \times 173392 + 0.75 \times 500 \times 1608 = 2944$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1500}{2944} = 0.510$, so $\alpha_n = 1.52$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{130.0}{210.5}\right)^{1.52} + \left(\frac{80.0}{138.2}\right)^{1.52} = 0.918 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 8-16 mm bars ($A_{sc} = 1608$ mm², $p = 0.92\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(16/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(350, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

Detailing

CROSS-SECTION 350 x 500
    +-------------------------+
    |   o        o        o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o        o        o   |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c

Answer: 350 × 500 mm column with 8-16 mm longitudinal bars ($p = 0.92\%$) and 8 mm ties @ 250 mm c/c.

Data and assumptions

• Column 300 mm (width $b$) × 450 mm (depth $D$); $P_u = 1500$ kN; $f_{ck} = 30$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 150 kN·m; about minor axis (y-y) $M_{uy}$ = 80 kN·m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3000/500 + 450/30 = 21.0$ mm (min. 20 mm); adopt 21.0 mm; $M_{ux,min} = P_u e = 31.5$ kN·m
• About y-y: $e_y = 3000/500 + 300/30 = 16.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 30.0$ kN·m

Design moments

• Design moments: $M_{ux} = 150.0$ kN·m, $M_{uy} = 80.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1500000}{30 \times 300 \times 450} = 0.370$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 30 \times 131073 + 0.75 \times 500 \times 3927 = 3242$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1500}{3242} = 0.463$, so $\alpha_n = 1.44$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{150.0}{238.6}\right)^{1.44} + \left(\frac{80.0}{134.3}\right)^{1.44} = 0.988 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 8-25 mm bars ($A_{sc} = 3927$ mm², $p = 2.91\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(300, 400, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 300 x 450
    +-------------------------+
    |                         |
    |    o       o       o    |
    |                         |
    |                         |
    |                         |
    |    o               o    |
    |                         |
    |                         |
    |                         |
    |    o       o       o    |
    |                         |
    +-------------------------+
   o = 25 mm bars, ties 8 mm @ 300 c/c

Answer: 300 × 450 mm column with 8-25 mm longitudinal bars ($p = 2.91\%$) and 8 mm ties @ 300 mm c/c.

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 changing the position of the neutral axis from very large (pure compression) to very small (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
   |     *
Pb |........* (balanced point)
   |         *
   |   tension*
   |   failure  *
   |             *
 0 +--------------*---> Mu
                 Mu (pure bending)

• Top point $P_{uz}$ (pure axial compression, $M = 0$); lowest point is pure bending.
• The balanced point $(M_b, P_b)$ is where concrete crushes ($\varepsilon_c = 0.0035$) and steel yields at the same time. It separates the two failure zones.
• Moment capacity first increases with axial load (below $P_b$) and then decreases (above $P_b$).
• Curves are plotted for different percentages of steel $p/f_{ck}$ and $d'/D$.
• 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 steel on the tension side has not yielded. It is sudden (brittle) failure and is on the upper part of the curve.
2. Balanced failure ($P_u = P_b$): concrete crushes and tension steel reaches yield simultaneously.
3. Tension failure (large eccentricity, $P_u < P_b$, $x_u < x_{u,max}$): the tension steel yields first, followed by crushing of concrete. It is gradual (ductile) failure and is on the lower part of the curve.
4. Pure compression ($e = 0$) is a limiting case; IS 456 limits to a minimum eccentricity $e_{min}$ so that this is not used.

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.

Data and assumptions

• Column 450 mm (width $b$) × 650 mm (depth $D$); $P_u = 1600$ kN; $f_{ck} = 30$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 90 kN·m; about minor axis (y-y) $M_{uy}$ = 55 kN·m.
• Moments are taken as: major-axis moments 90 kN·m (top) and 35 kN·m (bottom); minor-axis moments 55 kN·m (top) and 30 kN·m (bottom). The column bends in double curvature, so the smaller end moment is taken negative.
• Unsupported length for minimum eccentricity is taken equal to the larger effective length, 7 m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 7000/500 + 650/30 = 35.7$ mm (min. 20 mm); adopt 35.7 mm; $M_{ux,min} = P_u e = 57.1$ kN·m
• About y-y: $e_y = 7000/500 + 450/30 = 29.0$ mm (min. 20 mm); adopt 29.0 mm; $M_{uy,min} = 46.4$ kN·m

Design moments

• Major axis: $M_i = 0.6M_2 + 0.4M_1 = 0.6(90) + 0.4(-35) = 40.0$ kN·m, not less than $0.4M_2 = 36.0$; adopt 40.0 kN·m (cl. 39.7.1.1).
• Minor axis: $M_i = 0.6(55) + 0.4(-30) = 21.0$ kN·m, not less than $0.4M_2 = 22.0$; adopt 22.0 kN·m. Additional moment due to slenderness: $M_a = \dfrac{P_u D}{2000}\left(\dfrac{l_e}{D}\right)^2$, reduced by $k = \dfrac{P_{uz}-P_u}{P_{uz}-P_b} \le 1$.
• For the selected steel: $P_{uz} = 4821$ kN; $P_{by} = 1107$ kN, $k_y = 0.87$, $M_{ay} = 65.1$ kN·m.
• Design moments: $M_{ux} = 57.1$ kN·m, $M_{uy} = 87.1$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1600000}{30 \times 450 \times 650} = 0.182$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 30 \times 290087 + 0.75 \times 500 \times 2413 = 4821$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1600}{4821} = 0.332$, so $\alpha_n = 1.22$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{57.1}{492.3}\right)^{1.22} + \left(\frac{87.1}{325.0}\right)^{1.22} = 0.273 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-16 mm bars ($A_{sc} = 2413$ mm², $p = 0.82\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(16/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(450, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

Detailing

CROSS-SECTION 450 x 650
    +-------------------------+
    |  o      o     o      o  |
    |                         |
    |                         |
    |  o                   o  |
    |                         |
    |                         |
    |                         |
    |  o                   o  |
    |                         |
    |                         |
    |  o      o     o      o  |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c

Answer: 450 × 650 mm column with 12-16 mm longitudinal bars ($p = 0.82\%$) and 8 mm ties @ 250 mm c/c.

Data and assumptions

• Column 400 mm (width $b$) × 400 mm (depth $D$); $P_u = 1350$ kN; $f_{ck} = 25$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 150 kN·m; about minor axis (y-y) $M_{uy}$ = 100 kN·m.
• Both ends fully restrained for displacement and rotation: $l_e = 0.65L = 0.65 \times 3.4 = 2.21$ m (IS 456 Table 28).
• Section size is not given and is selected by trial.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3400/500 + 400/30 = 20.1$ mm (min. 20 mm); adopt 20.1 mm; $M_{ux,min} = P_u e = 27.2$ kN·m
• About y-y: $e_y = 3400/500 + 400/30 = 20.1$ mm (min. 20 mm); adopt 20.1 mm; $M_{uy,min} = 27.2$ kN·m

Design moments

• Design moments: $M_{ux} = 150.0$ kN·m, $M_{uy} = 100.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1350000}{25 \times 400 \times 400} = 0.338$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 156073 + 0.75 \times 500 \times 3927 = 3228$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1350}{3228} = 0.418$, so $\alpha_n = 1.36$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{150.0}{212.9}\right)^{1.36} + \left(\frac{100.0}{212.9}\right)^{1.36} = 0.977 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 8-25 mm bars ($A_{sc} = 3927$ mm², $p = 2.45\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 400, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 400 x 400
    +-------------------------+
    |                         |
    |   o        o        o   |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |   o        o        o   |
    |                         |
    +-------------------------+
   o = 25 mm bars, ties 8 mm @ 300 c/c

Answer: 400 × 400 mm column with 8-25 mm longitudinal bars ($p = 2.45\%$) and 8 mm ties @ 300 mm c/c.

Data and assumptions

• Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1600$ kN; $f_{ck} = 20$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 180 kN·m; about minor axis (y-y) $M_{uy}$ = 110 kN·m.
• Unsupported length for minimum eccentricity is taken equal to the larger effective length, 7 m.
• Single curvature: both end moments are of the same sign.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 7000/500 + 500/30 = 30.7$ mm (min. 20 mm); adopt 30.7 mm; $M_{ux,min} = P_u e = 49.1$ kN·m
• About y-y: $e_y = 7000/500 + 400/30 = 27.3$ mm (min. 20 mm); adopt 27.3 mm; $M_{uy,min} = 43.7$ kN·m

Design moments

• Major axis: $M_i = 0.6M_2 + 0.4M_1 = 0.6(180) + 0.4(100) = 148.0$ kN·m, not less than $0.4M_2 = 72.0$; adopt 148.0 kN·m (cl. 39.7.1.1).
• Minor axis: $M_i = 0.6(110) + 0.4(50) = 86.0$ kN·m, not less than $0.4M_2 = 44.0$; adopt 86.0 kN·m. Additional moment due to slenderness: $M_a = \dfrac{P_u D}{2000}\left(\dfrac{l_e}{D}\right)^2$, reduced by $k = \dfrac{P_{uz}-P_u}{P_{uz}-P_b} \le 1$.
• For the selected steel: $P_{uz} = 3580$ kN, $P_{bx} = 331$ kN, $k_x = 0.61$, $M_{ax} = 47.8$ kN·m; $P_{by} = 247$ kN, $k_y = 0.59$, $M_{ay} = 42.8$ kN·m.
• Design moments: $M_{ux} = 195.8$ kN·m, $M_{uy} = 128.8$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1600000}{20 \times 400 \times 500} = 0.400$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 194110 + 0.75 \times 415 \times 5890 = 3580$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1600}{3580} = 0.447$, so $\alpha_n = 1.41$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{195.8}{325.3}\right)^{1.41} + \left(\frac{128.8}{241.4}\right)^{1.41} = 0.900 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-25 mm bars ($A_{sc} = 5890$ mm², $p = 2.95\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 400, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 400 x 500
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 25 mm bars, ties 8 mm @ 300 c/c

Answer: 400 × 500 mm column with 12-25 mm longitudinal bars ($p = 2.95\%$) and 8 mm ties @ 300 mm c/c.

Data and assumptions

• Column 350 mm (width $b$) × 550 mm (depth $D$); $P_u = 900$ kN; $f_{ck} = 25$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moment about major axis: $M_{ux}$ = 200 kN·m.
• Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3000/500 + 550/30 = 24.3$ mm (min. 20 mm); adopt 24.3 mm; $M_{ux,min} = P_u e = 21.9$ kN·m
• About y-y: $e_y = 3000/500 + 350/30 = 17.7$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 18.0$ kN·m

Design moments

• Design moments: $M_{ux} = 200.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{900000}{25 \times 350 \times 550} = 0.187$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

Check

$M_{ux} = 200.0$ kN·m $\le M_{ux1} = 231.7$ kN·m. Minimum eccentricity about the minor axis: $P_u e_y = 18.0$ kN·m $\le M_{uy1} = 137.4$ kN·m. Both are satisfied; the section is safe.

Provide 8-16 mm bars ($A_{sc} = 1608$ mm², $p = 0.84\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(16/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(350, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

Detailing

CROSS-SECTION 350 x 550
    +-------------------------+
    |   o        o        o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o        o        o   |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c

Answer: 350 × 550 mm column with 8-16 mm longitudinal bars ($p = 0.84\%$) and 8 mm ties @ 250 mm c/c.