Civil Engineering Reference
In-Depth Information
Let
σ
a
be the mean axial stress due to
P
taken over the complete area,
A
, of the beam
section. Then
P
=
σ
a
A
Substituting for
P
in Eq. (18.21)
Z
P
−
A
2
4
t
w
σ
a
σ
Y
M
P,R
=
σ
Y
(18.22)
Thus the reduced plastic section modulus may be expressed in the form
Kn
2
Z
P,R
=
Z
P
−
(18.23)
where
K
is a constant that depends upon the geometry of the beam section and
n
is
the ratio of the mean axial stress to the yield stress of the material of the beam.
Equations (18.22) and (18.23) are applicable as long as the neutral axis lies in the web
of the beam section. In the rare case when this is not so, reference should be made
to advanced texts on structural steel design. In addition the design of beams carrying
compressive loads is influenced by considerations of local and overall instability, as we
shall see in Chapter 21.
18.3 P
LASTIC
A
NALYSIS OF
F
RAMES
The plastic analysis of frames is carried out in a very similar manner to that for beams
in that possible collapse mechanisms are identified and the principle of virtual work
used to determine the collapse loads. A complication does arise, however, in that
frames, even though two-dimensional, can possess collapse mechanisms which involve
both
beam
and
sway
mechanisms since, as we saw in Section 16.10 in the moment
distribution analysis of portal frames, sway is produced by any asymmetry of the loading
or frame. Initially we shall illustrate the method by a comparatively simple example.
E
XAMPLE
18.8
Determine the value of the load
W
required to cause collapse of
the frame shown in Fig. 18.16(a) if the plastic moment of all members of the frame is
200 kNm. Calculate also the support reactions at collapse.
We note that the frame and loading are unsymmetrical so that sway occurs. The bending
moment diagram for the frame takes the form shown in Fig. 18.16(b) so that there are
three possible collapse mechanisms as shown in Fig. 18.17.
In Fig. 18.17(a) the horizontal member BCD has collapsed with plastic hinges forming
at B, C and D; this is termed a
beam mechanism
. In Fig. 18.17(b) the frame has
swayed with hinges forming at A, B, D and E; this, for obvious reasons, is called a
sway mechanism
. Fig. 18.17(c) shows a
combined mechanism
which incorporates both
