Built-up laced column, 10 m, 1000 kN

Given: factored axial load $P=1000$ kN, length 10 m, held in position but not restrained in direction at both ends → $K=1.0$, $KL=10$ m. Two channels back to back with single bolted lacing; E250 ($f_y=250$, $f_u=410$); bolts M20 grade 4.6. Built-up members belong to buckling class c (IS 800:2007 Table 10, $\alpha=0.49$).

1. Trial section: 2 ISMC 300

$A=2\times4564=9128$ mm², $I_{xx}=2\times6362.6\times10^4$ mm⁴, $r_{xx}=118.1$ mm, $C_{yy}=23.6$ mm, $b_f=90$ mm.

• $\lambda_{xx}=\dfrac{KL}{r_{xx}}=\dfrac{10000}{118.1}=84.7$
• Effective slenderness of laced column $=1.05\lambda=88.9$ (cl. 7.6.1.4)
• $f_{cc}=\dfrac{\pi^2E}{\lambda^2}$, $\lambda_n=\sqrt{f_y/f_{cc}}$, $\phi=0.5[1+0.49(\lambda_n-0.2)+\lambda_n^2]$, $f_{cd}=\dfrac{f_y/\gamma_{m0}}{\phi+\sqrt{\phi^2-\lambda_n^2}}=122.6$ N/mm²
• $P_d=A f_{cd}=9128\times122.6\times10^{-3}=1119$ kN $>1000$ ✓

2. Spacing of channels (so that $I_{yy}\ge I_{xx}$)

$$\left(C_{yy}+\frac{s}{2}\right)^2\ge\frac{I_{xx}-I_{yy}}{A_{ch}}\Rightarrow C_{yy}+\frac{s}{2}\ge115.2\Rightarrow s\ge183\ \text{mm}$$

Provide a clear gap $s=190$ mm between the webs. Then $I_{yy}=2[I_{yy,ch}+A_{ch}(C_{yy}+s/2)^2]=13461\times10^4$ mm⁴, $r_{yy}=121.4$ mm, $\lambda_{yy}=82.3\ (<\lambda_{xx})$, $1.05\lambda_{yy}=86.5$, $P_{dy}=1153$ kN ✓.

3. Lacing (single, bolted)

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

• Distance between bolt lines (at the middle of the flanges) $S=s+b_f=190+90=280$ mm
• Transverse shear $V=2.5\%P=25.0$ kN; two lacing planes: $V_p=12.50$ kN per plane
• Lacing at $\theta=45°$ (limits 40°-70°): force $F=\dfrac{V_p}{\sin\theta}=17.68$ kN
• Length of bar between end bolts $l=\dfrac{S}{\sin\theta}=396$ mm
• Thickness $t\ge\dfrac{l}{40}=10$ mm (single lacing); width $\ge3d=60$ mm → flat 60 × 10 mm
• Slenderness $\lambda=\dfrac{l}{t/\sqrt{12}}=137<145$ ✓; $f_{cd}=68$ N/mm² → compressive strength $=41.0$ kN $>F$ ✓; tension $T_{dn}=\dfrac{0.9(60-22)t f_u}{1.25}=112.2$ kN $>F$ ✓
• Bolt connection: $V_{dsb}=45.3$ kN (single shear), bearing on 10 mm $=99.4$ kN → 1 M20 bolt at each end
• Spacing of lacing points on one channel $=\dfrac{2S}{\tan\theta}=560$ mm; slenderness of the channel between them $=\dfrac{560}{26.1}=21.5\le\min(50,\,0.7\times89)$ ✓ (cl. 7.6.3).

4. Tie plates

The lacing ends at tie (batten) plates at each end: length $\ge S$ + edges $=360$ mm, depth $\ge S=280$ mm for a plate at the end, thickness $\ge S/50=6$ mm: tie plate 360 × 280 × 6 mm, connected by 2 M20 bolts to each channel.

Answer: 2 ISMC 300 back to back, clear gap 190 mm, single lacing of 60 × 10 mm flats at 45° with M20 bolts, tie plates at the ends.