Civil Engineering Reference
In-Depth Information
Figure 14.13
beam plus a beam with end moments. Thus the final moment diagram for the fixed-end
beam equals the moment diagram if the beam had been simply supported plus the end mo-
ment diagram.
For the beam under consideration, the value of
M
n
can be calculated as follows (see
Figure 14.13):
2
w
n
2
M
n
8
2
w
n
M
n
16
The propped beam of Figure 14.14, which supports a concentrated load, is presented
as a second illustration of plastic analysis. It is desired to determine the value of
P
n
, the
theoretical ultimate load the beam can support before collapse. The maximum moment in
this beam in the elastic range occurs at the fixed end, as shown in the figure. If the magni-
tude of the concentrated load is increased, the moments in the beam will increase propor-
tionately until a plastic moment is eventually developed at some point. This point will be
at the fixed end, where the elastic moment diagram has its largest ordinate.
After this plastic hinge is formed, the beam will act as though it is simply supported
insofar as load increases are concerned, because it will have a plastic hinge at the left end
and a real hinge at the right end. An increase in the magnitude of the load
P
will not in-
crease the moment at the left end but will increase the moment out in the beam, as it
would in a simple beam. The increasing simple beam moment is indicated by the dashed
line in Figure 14.14(c). Eventually, the moment at the concentrated load will each
M
n
and
a mechanism will form, consisting of two plastic hinges and one real hinge, as shown in
Figure 14.14(d).
The value of the theoretical maximum concentrated load
P
n
that the beam can support
can be determined by taking moments to the right or left of the load. Figure 14.14(e)
shows the beam reactions for the conditions existing just before collapse. Moments are
taken to the right of the load as follows:
P
n
2
M
n
20
M
n
10
P
n
0.3
M
n
The subject of plastic analysis can be continued for different types of structures and
loadings, as described in several textbooks on structural analysis or steel design.
2
The
2
McCormac, J. C., and Nelson, J. K., Jr., 2003,
Structural Steel Design: LRFD Method
, 3rd ed. (Upper Saddle
River, NJ: Prentice Hall), pp. 215-231.
