The Design of Modern Steel Bridges - TEDI

98 The Design of Modern Steel Bridges
For the I-section the contribution of the web to the minor axis moment of
inertia Iy is negligible; the critical bending moment M cr can be considered to be
PE (hb= ffiffiffi p
a),
where PE is the Euler critical load of the compression flange.
Another way of expressing equation (5.4) is
Ah b
Mcr ¼ sE pffiffiffi ð5:5aÞ
2 a
where
or sEZe
or sEZp
2Af þ Aw
2Af þ 1
3 Aw
2Af þ Aw
2Af þ 1
2 Aw
s E ¼ p 2 E/(L e/r y) 2 , i.e. the Euler stress of the whole beam
ry ¼ the radius of gyration of the beam about its minor axis
Z e ¼ elastic section modulus
Zp ¼ plastic section modulus
A f ¼ area of each flange
A w ¼ area of web
A ¼ total area ¼ 2A f þ A w.
!
!
b
pffiffiffi ð5:5bÞ
a
b
pffiffiffi ð5:5cÞ
a
Expression (5.3) for b may be simplified by making some approximations
about the geometric properties of equal-flange I-sections; for example, if it is
assumed that (i) the area of the web equals that of each flange, (ii) the web
contribution to J is 0.6 times that of each flange, and (iii) h ¼ 0.95D, D being
the overall depth of the beam, then
Iy ¼ 1
6 B3T Taking E ¼ 2.6G leads to
J ¼ 0:87BT 3
ð0:95DÞ
Iw ¼ Iy
2
¼
4
B3D2T 26:6
r 2 y
¼ B2
18
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ 1 þ 1
s
20
Le T
D
2
ry
ð5:6Þ