In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 162

M 1 M

M y 1

y 1

y 1 ( v 1 = 0) M y

M y 1

y

y M z

M z 1

Plane of
moment Principal

planes

M

z

z 1 z

Plane of
deflection

z 1

(a) Deflection in a non-principal plane (b) Principal plane moments M y , M z

Figure 5.7 Bending in a non-principal plane.

q z ( )

+

x

M y + ve

C

y

z,w

q y ( )

+

(b) Bending M y in xz plane

z

q y ( ) +

x

q z ( ) +

~

M z ve

q

y, v

(a) Section and loading (c) Bending M z in xy plane

Figure 5.8 Biaxial bending of a zed beam.

inwhich α istheanglebetweenthe y 1 , z 1 axesandtheprincipal y , z axes,asshown
in Figure 5.7b.The bending stresses can then be determined from

σ = M y z
I y

− M z y

I z ,

(5.12)

in which I y , I z are the principal second moments of area. If the values of α , I y , I z
are unknown, they can be determined from I y 1 , I z 1 , I y 1 z 1 as shown in Section 5.9.
Care needs to be taken to ensure that the correct signs are used for M y and M z
in equation 5.12, as well as for y and z . For example, if the zed section shown in
Figure 5.8a is simply supported at both ends with a uniformly distributed load q
acting in the plane of the web, then its principal plane components q y , q z cause
positive bending M y and negative bending M z , as indicated in Figure 5.8b and c.