Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 329

7.7 Worked examples

7.7.1 Example 1 - approximating the maximum elastic
moment

Problem. Use the approximations of Figures 7.5 and 7.6 to determine the maxi-
mum moments M max , y , M max , z for the 254 × 146 UB 37 beam-columns shown in
Figure 7.19d and c.
Solution for M max , y for the beam-column of Figure 7.19d.
Using Figure 7.5, k cr = 1.0, γ m = 1.0, γ n = 1.0, γ s = 0.18
M y = 20 × 9 / 4 = 45 kNm.
N cr , y = π 2 × 210 000 × 5537 × 10 4 /( 1.0 × 9000 ) 2 N = 1417kN.
Using equation 7.10, δ y = 1.0 × ( 1 − 0.18 × 200 / 1417 )
( 1 − 1.0 × 200 / 1417 )

= 1.135.

Using equation 7.9, M max , y = 1.135 × 45 = 51.1 kNm.
Solution for M max , z for the beam-column of Figure 7.19c.
Using Figure 7.6, k cr = 1.0, γ m = 1.0, γ n = 0.49, γ s = 0.18
M z = 3.2 × 4.5 2 / 8 = 8.1 kNm.
N cr , z = π 2 × 210 000 × 571 × 10 4 /( 1.0 × 4500 ) 2 N = 584.4 kN.
Using equation 7.10, δ z = 1.0 × ( 1 − 0.18 × 200 / 584.4 )
( 1 − 0.49 × 200 / 584.4 )

= 1.127.

Using equation 7.9, M max , z = 1.127 × 8.1 = 9.1 kNm.

20 kN

h = 256.0 mm
b f = 146.4 mm
t f = 10.9 mm
t w = 6.3 mm
r = 7.6 mm
A = 47.2 cm 2
I t = 15.3 cm 4
I w = 0.0857 dm 6
I y = 5537 cm 4
i y = 10.8 cm
W pl,y = 483 cm 3
W el,y = 433 cm 3
I z = 571 cm 4
i z = 3.48 cm
W pl,z = 119 cm 3
W el,z = 78.0 cm 3

200 kN x

4500 4500

(b) Example 2

z

3.2 kN/m

y 200 kN x

4500 4500

(c)Examples
1, 3, 5

y

z
(a) 254 × 146 UB 37

20 kN

v = = 0

200 kN

x

4500 4500

(d) Examples
1, 4, 5

z

Figure 7.19 Examples 1-5.