Civil Engineering Reference
In-Depth Information
Fully plastic
zone
s
Y
d
e
Elastic and
plastic neutral axis
d
s
Y
F
IGURE
18.8
Plastic
bending of a
rectangular-section beam
b
(a)
(b)
as an asymptote when, theoretically, the curvature is infinite at the collapse load. From
Eq. (18.7) we see that when
M
f
, the shape factor. Clearly the
equation of the non-linear portion of the moment-curvature diagram depends upon
the particular cross section being considered.
=
M
P
, the ratio
M
/
M
Y
=
Suppose a beam of rectangular cross section is subjected to a bending moment which
produces fully plastic zones in the outer portions of the section (Fig. 18.8(a)); the
depth of the elastic core is
d
e
. The total bending moment,
M
, corresponding to the
stress distribution of Fig. 18.8(b) is given by
d
2
+
2
σ
Y
b
1
d
e
)
1
2
d
e
2
2
σ
Y
2
b
d
e
2
3
d
e
2
M
=
2
(
d
−
+
2
which simplifies to
3
3
σ
Y
bd
2
12
d
e
d
2
d
e
d
2
M
Y
2
M
=
−
=
−
(18.11)
Note that when
d
e
=
d
,
M
=
M
Y
and when
d
e
=
0,
M
=
3
M
Y
/
2
=
M
P
as derived in
Ex. 18.1.
The curvature of the beam at the section shown may be found using Eq. (9.2) and
applying it to a point on the outer edge of the elastic core. Thus
E
d
e
2
R
σ
Y
=
or
1
R
=
2
σ
Y
Ed
e
k
=
(18.12)
The curvature of the beam at yield is obtained from Eq. (18.9), i.e.
M
Y
EI
=
2
σ
Y
Ed
k
Y
=
(18.13)
Combining Eqs (18.12) and (18.13) we obtain
k
k
Y
=
d
d
e
(18.14)
