Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 284

(4) Using M zx 12 = M zx 23 = 111.2kNm from Section 6.15.2,

M cr 12 = 2.448 × 111.2 = 272.3kNm.
Q s 12 = 70 × 272.3 / 122.5 = 155.6kN.
M cr 23 = 1.75 × 111.2 = 194.6kNm.
Q s 23 = 70 × 194.6 / 122.5 = 111.2kN.

(5) Q s 23 < Q s 12 and so 23 is the critical segment.
(6) α r 12 = ( 3 × 210000 × 571 × 10 4 / 4500 ) × ( 1 − 111.2 / 155.6 )
= 227.9 × 10 6 Nmm.
(7) α 23 = 2 × 210000 × 571 × 10 4 / 4500 = 532.9 × 10 6 Nmm.
(8) k 2 = 532.9 × 10 6 /( 0.5 × 227.9 × 10 6 + 532.9 × 10 6 ) = 0.824
k 3 = 1.0 (zero restraint at 3).
(9) Using Figure 3.21a, k cr 23 = 0.93, L cr 23 = 0.93 × 4500 = 4185mm.
(10) Using equation 6.39,

π 2 × 210000 × 571 × 10 4
4185 2

M zx =

Nmm

81000 × 15.3 × 10 4 + π 2 × 210000 × 0.0857 × 10 12
4185 2

×

= 123.4kNm

and M cr 23 = 1.75 × 123.4 = 215.9kNm.

(Using the computer program PRFELB [18] leads to M cr 23 = 237.9kNm.)
Member resistance.
M c , Rd = 132.8kNm as in Section 6.15.2

Using equation 6.25, λ LT = √ ( 132.8 / 215.9 ) = 0.784

h / b = 256 / 146.4 = 1.75 < 2

Using the EC3 less conservative method with β = 0.75, λ LT ,0 = 0.4 (Clause
6.3.2.3) and α LT = 0.34 (Tables 6.5 and 6.3), and equation 6.27,

Φ LT = 0.5 { 1 + 0.34 ( 0.784 − 0.4 ) + 0.75 × 0.784 2 }= 0.796

Using equation 6.26,

M b , Rd = 132.8 / { 0.796 + √ ( 0.796 2 − 0.75 × 0.784 2 ) } kNm = 105.6kNm.