Civil Engineering Reference
In-Depth Information
If the bending moment is further increased, the strain at the extremity
y
1
of the section
increases and exceeds the yield strain,
ε
Y
. However, due to plastic yielding the stress
remains constant and equal to
σ
Y
as shown in the idealized stress-strain curve of
Fig. 18.1. At some further value of
M
the stress at the lower extremity of the section also
reaches the yield stress,
σ
Y
(Fig. 18.3(c)). Subsequent increases in bending moment
cause the regions of plasticity at the extremities of the beam section to extend inwards,
producing a situation similar to that shown in Fig. 18.3(d); at this stage the central
portion or 'core' of the beam section remains elastic while the outer portions are
plastic. Finally, with further increases in bending moment the elastic core is reduced
to a negligible size and the beam section is more or less completely plastic. Then, for
all practical purposes the beam has reached its ultimate moment resisting capacity;
the value of bending moment at this stage is known as the
plastic moment, M
P
, of the
beam. The stress distribution corresponding to this moment may be idealized into two
rectangular portions as shown in Fig. 18.3(e).
The problem now, therefore, is to determine the plastic moment,
M
P
. First, however,
we must investigate the position of the neutral axis of the beam section when the latter
is in its fully plastic state. One of the conditions used in establishing that the elastic
neutral axis coincides with the centroid of a beam section was that stress is directly
proportional to strain (Eq. (9.2)). It is clear that this is no longer the case for the stress
distributions of Figs 18.3(c), (d) and (e). In Fig. 18.3(e) the beam section above the
plastic neutral axis
is subjected to a uniformcompressive stress,
σ
Y
, while below the neu-
tral axis the stress is tensile and also equal to
σ
Y
. Suppose that the area of the beam
section below the plastic neutral axis is
A
2
, and that above,
A
1
(Fig. 18.4(a)). Since
M
P
is a pure bending moment the total direct load on the beam section must be zero.
Thus from Fig. 18.4
σ
Y
A
1
=
σ
Y
A
2
so that
A
1
=
A
2
(18.2)
s
Y
Area,
A
1
C
1
y
1
Plastic neutral
axis
y
2
C
2
Area,
A
2
F
IGURE
18.4
Position of the
plastic neutral axis
in a beam section
s
Y
(a)
(b)
