In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 176

Stress

Strain-hardened
E st

Plastic

f y

Rigid plastic
assumption

Elastic E

0 y st

Strain

Figure 5.24 Idealised stress-strain relationships for structural steel.

yielded at the fully plastic moment

M p = f y W pl , (5.33)

where W pl is the plastic section modulus. Methods of calculating the fully plastic
moment M p are discussed in Section 5.11.1, and worked examples are given in
Sections 5.12.12 and 5.12.13. For solid rectangular sections, the shape factor
W pl / W el is 1.5, but for rolled I-sections, W pl / W el varies between 1.1 and 1.2
approximately.Values of W pl for hot-rolled I-section members are given in [9].
In real beams, strain-hardening commences just before M p is reached, and the
real moment-curvature relationship rises above the fully plastic limit of M p ,as
showninFigure5.2d.Ontheotherhand,highshearforcescausesmallreductions
in M p below f y W pl , due principally to reductions in the plastic bending capacity
of the web.This effect is discussed in Section 4.5.2.
The approximate approach of the moment-curvature relationship shown in
Figure 5.2d to the fully plastic limit forms the basis of the simple rigid-plastic
assumption for which the basic stress-strain relationship is replaced by the rect-
angular block shown by the dashed line in Figure 5.24. Thus elastic strains
are completely ignored, as are the increased stresses due to strain-hardening.
This assumption ignores the curvature of any elastic and elastic-plastic regions
( M < M p ) , and assumes that the curvature becomes infinite at any point where
M = M p .
The consequences of the rigid-plastic assumption on the theoretical behaviour
of a simply supported beam with a central concentrated load are shown in
Figures 5.25 and 5.26. In the real beam shown in Figure 5.25a, there is a finite
length of the beam which is elastic-plastic and in which the curvatures are large,
while the remaining portions are elastic, and have small curvatures. However,
according to the rigid-plastic assumption, the curvature becomes infinite at mid-
span when this section becomes fully plastic ( M = M p ) , while the two halves of
the beam have zero curvature and remain straight, as shown in Figure 5.25b.The