Civil Engineering Reference
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x
C
+ve
M
y
-ve d
2
w
/d
x
2
z,w
(b) Curvature in
xz
plane
M
y
x
C
C
M
y
y
x
z
+ve
M
z
+ve d
2
v
/d
x
y, v
M
z
M
z
(c) Curvature in
xy
plane
(a) Positive bending actions
Figure 5.5
Bending sign conventions.
σ
max
=
M
z
/
W
el
,
zL
and
(5.7)
σ
max
=−
M
z
/
W
el
,
zR
in which
W
el
,
zL
=−
I
z
/
y
L
and
W
el
,
zR
=
I
z
/
y
R
are the elastic section moduli for
bending about the
z
axis.
Values of
I
y
,
I
z
,
W
el
,
z
,
W
el
,
z
for hot-rolled steel sections are given in [9], while
valuesforothersectionscanbecalculatedasindicatedinSection5.9orinstandard
textbooks [1, 2]. Expressions for the properties of some thin-walled sections are
given in Figure 5.6 [10].When a section has local holes, or excessive widths (see
Section 4.2.2.2), these properties may need to be reduced accordingly.
Worked examples of the calculation of cross-section properties are given in
Sections 5.12.1-5.12.4.
5.3.2 Biaxial bending
When a beam deflects only in a plane
xz
1
which is not a principal plane (see
Figure 5.7a), so that its curvature d
2
v
1
/d
x
2
in the perpendicular
xy
1
plane is zero,
then the bending stresses
σ
=−
Ez
1
d
2
w
1
/
d
x
2
(5.8)
have moment resultants
M
y
1
=−
EI
y
1
d
2
w
1
/
d
x
2
and
(5.9)
M
z
1
=−
EI
y
1
z
1
d
2
w
1
/
d
x
2
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