In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 190

usedforcranegantrygirders.Avalueof H / 300hasbeenusedforhorizontalstorey
drift in buildings under wind load.
Aworked example of the serviceability design is given in Section 5.12.18.

5.8 Appendix - bending stresses in elastic beams

5.8.1 Bending in a principal plane

When a beam bends in the xz principal plane, plane cross-sections rotate as
shown in Figure 5.4, so that sections δ x apart become inclined to each other at
− ( d 2 w / d x 2 ) δ x , where w is the deflection in the z principal direction as shown
in Figure 5.5, and δ x the length along the centroidal axis between the two cross-
sections. The length between the two cross-sections at a distance z from the axis
is greater than δ x by z ( − d 2 w / d x 2 ) δ x , so that the longitudinal strain is

ε =− z d 2 w
d x 2

The corresponding tensile stress σ = E ε is

σ =− Ez d 2 w
d x 2

(5.50)

which has the stress resultants

N = σ d A = 0

A

since

A z d A = 0 for centroidal axes (see Section 5.9), and

M y = σ z d A

A

whence

M y =− EI y d 2 w
d x 2

(5.2)

since

A z 2 d A = I y (see Section 5.9). If this is substituted into equation 5.50, then
the bending tensile stress σ is obtained as

σ = M y z

I y .

(5.3)