Civil Engineering Reference
In-Depth Information
Equation (18.6) may be interpreted as the first moment, about the plastic neutral axis,
of the area above the plastic neutral axis plus the first moment of the area below the
plastic neutral axis. Hence
133 800mm
3
Z
P
=
150
×
9
.
4
×
4
.
7
+
150
×
0
.
6
×
0
.
3
+
190
×
7
×
95
.
6
=
The shape factor
f
is, from Eq. (18.7)
M
P
M
Y
=
Z
P
Z
e
=
133 800
75 000
=
f
=
1
.
78
MOMENT-CURVATURE RELATIONSHIPS
From Eq. (9.8) we see that the curvature
k
of a beam subjected to elastic bending is
given by
1
R
=
M
EI
k
=
(18.8)
At yield, when
M
is equal to the yield moment,
M
Y
M
Y
EI
k
Y
=
(18.9)
The moment-curvature relationship for a beam in the linear elastic range may
therefore be expressed in non-dimensional form by combining Eqs (18.8) and
(18.9), i.e.
M
M
Y
=
k
k
Y
(18.10)
This relationship is represented by the linear portion of the moment-curvature dia-
gram shown in Fig. 18.7. When the bending moment is greater than
M
Y
part of the
beam becomes fully plastic and the moment-curvature relationship is non-linear. As
the plastic region in the beam section extends inwards towards the neutral axis the
curve becomes flatter as rapid increases in curvature are produced by small increases in
moment. Finally, themoment-curvature curve approaches the horizontal line
M
=
M
P
M/M
Y
M
M
P
f
M
M
Y
1
F
IGURE
18.7
Moment-curvature diagram
for a beam
k
/
k
Y
1
