Built-up column, two channels toe-to-toe, 1500 kN

Given/assumptions: factored load $P_u=1500$ kN; effective length $KL=10$ m (given); two channels toe to toe with single lacing and welded connection. "Fe 415" is read as ordinary structural steel with $f_y=250$ N/mm², $f_u=410$ N/mm² (IS 2062 E250); if $f_y=415$ were intended, redo the design with that value. Shop fillet welds.

1. Section: 2 ISMC 350 toe to toe (flanges facing, webs outward)

$A=2\times5366=10732$ mm², $I_{xx}=2\times10008.0\times10^4$ mm⁴, $r_{xx}=136.6$ mm, $C_{yy}=24.4$ mm, $b_f=100$ mm. Built-up members use buckling class c (IS 800:2007 Table 10, $\alpha=0.49$).

• $\lambda_{xx}=\dfrac{KL}{r_{xx}}=\dfrac{10000}{136.6}=73.2$
• Laced column: $\lambda_e=1.05\lambda=76.9$ (cl. 7.6.1.4)
• $f_{cd}=\dfrac{f_y/\gamma_{m0}}{\phi+\sqrt{\phi^2-\lambda_n^2}}$ with $\phi=0.5[1+0.49(\lambda_n-0.2)+\lambda_n^2]$, $\lambda_n=\sqrt{f_y/f_{cc}}$, $f_{cc}=\pi^2E/\lambda_e^2$ → $f_{cd}=141.2$ N/mm²
• $P_d=A f_{cd}=1515$ kN $\ge P_u=1500$ kN ✓

2. Clear gap for equal stiffness

The centroid of each section is at $x_c=75.6+s/2$ from the column axis. For $I_{yy}\ge I_{xx}$:

$$x_c\ge\sqrt{\frac{I_{xx}-I_{yy,ch}}{A_{ch}}}=133.6\ \text{mm}\ \Rightarrow\ s\ge116\ \text{mm}$$

Provide $s=120$ mm. Then $I_{yy}=2[I_{yy,ch}+A_{ch}x_c^2]=20595\times10^4$ mm⁴, $r_{yy}=138.5$ mm, $\lambda_{yy}=72.2$, $1.05\lambda_{yy}=75.8$, $P_{dy}=1534$ kN $\ge P_u$ ✓.

3. Design of lacing (single lacing, two planes)

   |\     /|
   | \   / |    theta = 45 deg to the axis
   |  \ /  |    S = distance between connection lines
   |  / \  |
   | /   \ |

• Distance between connection lines $S=s+b_f=120+100=220$ mm
• Transverse shear $V=2.5\%\,P_u=37.5$ kN; per lacing plane $V_p=18.75$ kN
• Force in lacing bar $F=\dfrac{V_p}{\sin\theta}=26.52$ kN
• Length of bar $l=\dfrac{S}{\sin\theta}=311$ mm; effective length for welded lacing $=0.7l=218$ mm
• $t\ge l_e/40=5.5$ mm; a 6 mm flat fails in compression (23.4 kN $<F$), so adopt 8 mm; width 50 mm → flat 50 × 8 mm
• $\lambda=\dfrac{l_e}{t/\sqrt{12}}=94<145$ ✓; $f_{cd}=115$ N/mm², $P_c=45.9$ kN $>F$ ✓
• Weld: 5 mm shop fillet; strength $=\dfrac{410}{\sqrt3\times1.25}\times0.7\times5=662.8$ N/mm → length required $=\dfrac{F}{q}=40$ mm; provide 55 mm of weld at each end (not less than the bar width and $4s$), returned round the ends
• Lacing angle $\theta=45°$ lies between 40° and 70° ✓
• Spacing of lacing points on one section $=\dfrac{2S}{\tan\theta}=440$ mm; slenderness of the section between them $=\dfrac{440}{28.3}=15.5\le\min(50,\ 0.7\lambda_{max})$ ✓ (cl. 7.6.3)

4. Tie plates

Tie (end) plates at both ends of the lacing system: length $\ge S+$ edges $=300$ mm, depth $\ge S=220$ mm, thickness $\ge S/50=6$ mm → 300 × 220 × 6 mm at each end, fillet-welded to each section.

Answer: 2 ISMC 350 toe to toe (flanges facing, webs outward), clear gap 120 mm, single lacing of 50 × 8 mm flats at 45° (welded), tie plates 300 × 220 × 6 mm at the ends.