Civil Engineering Reference
In-Depth Information
equation5.67areadditive.Theplasticsectionmodulus
W
pl
,
y
=
M
p
,
y
/
f
y
isobtained
from equation 5.67 as
W
pl
,
y
=
A
n
z
n
−
A
n
z
n
.
(5.68)
T
C
A worked example of the determination of the full plastic moment of a doubly
symmetric I-section is given in Section 5.12.12.
Thefullyplasticstressdistributioninamonosymmetricsectionbentinaplane
of symmetry is not antisymmetric, as can be seen in Figure 5.37b for a channel
section example. The plastic neutral axis again divides the cross-section into two
equal areas so that there is no axial force resultant, but the neutral axis no longer
passes through the centroid of the cross-section. The fully plastic moment
M
p
,
y
and the plastic section modulus
W
pl
,
y
are given by equations 5.67 and 5.68. For
convenience,anysuitableoriginmaybeusedforcomputing
z
n
.Aworkedexample
ofthedeterminationofthefullplasticmomentofamonosymmetrictee-sectionis
given in Section 5.12.13.
Thefullyplasticstressdistributioninanasymmetricsectionisalsonotantisym-
metric, as can be seen in Figure 5.37c for an angle section example. The plastic
neutral axis again divides the cross-section into two equal areas.The direction of
the plastic neutral axis is perpendicular to the plane in which the moment resul-
tant acts. The full plastic moment is given by the vector addition of the moment
resultants about any convenient
y
1
,
z
1
axes, so that
M
p
=
√
(
M
p
,
y
1
+
M
p
,
z
1
)
(5.69)
where
M
p
,
y
1
is given by an equation similar to equation 5.67, and
M
p
,
z
1
is
given by
M
p
,
z
1
=−
f
y
A
n
y
1
n
+
f
y
A
n
y
1
n
.
(5.70)
T
C
The inclination
α
of the plastic neutral axis to the
y
1
axis (and of the plane of the
full plastic moment to the
zx
plane) can be obtained from
tan
α
=
M
p
,
z
1
/
M
p
,
y
1
.
(5.71)
5.11.2 Built-in beam
5.11.2.1 General
Thebuilt-inbeamshowninFigure5.30ahastworedundancies,andsothreeplastic
hinges must form before it can be reduced to a collapse mechanism. In this case
theonlypossibleplastichingelocationsareatthesupportpoints1and4andatthe
loadpoints2and3,andsotherearetwopossiblemechanisms,onewithahingeat
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