Civil Engineering Reference
In-Depth Information
We can now use the lower bound theorem to check that we have obtained the critical
mechanism and thereby the critical load. The internal moment at A at collapse is
hogging and equal to
M
P
. Then, taking moments about A
w
L
2
2
R
B
L
−
=−
M
P
which gives
4
.
83
M
P
L
R
B
=
Similarly, taking moments about B gives
6
.
83
M
P
L
R
A
=
Summation of
R
A
and
R
B
gives 11.66
M
P
/
L
wL
so that vertical equilibrium is
satisfied. Further, considering moments of forces to the right of C about C we have
=
w
0
.
414
L
2
2
M
C
=
R
B
(0
.
414
L
)
−
Substituting for
R
B
and
w
from the above gives
M
C
=
M
P
. The same result is obtained
by considering moments about C of forces to the left of C. The load therefore satisfies
both vertical and moment equilibrium.
The bending moment at any distance
x
1
, say, from B is given by
w
x
1
2
M
=
R
B
x
1
−
Then
d
M
d
x
1
=
R
B
−
wx
1
=
0
so that a maximum occurs when
x
1
=
R
B
/
w
. Substituting for
R
B
,
x
1
and
w
in the expres-
sion for
M
gives
M
M
P
so that the yield criterion is satisfied. We conclude, therefore,
that the mechanism of Fig. 18.14(b) is the critical mechanism.
=
PLASTIC DESIGN OF BEAMS
It is now clear that the essential difference between the plastic and elastic methods of
design is that the former produces a structure having a more or less uniform factor of
safety against collapse of all its components, whereas the latter produces a uniform
factor of safety against yielding. The former method in fact gives an indication of
the true factor of safety against collapse of the structure which may occur at loads
only marginally greater than the yield load, depending on the cross sections used. For
example, a rectangular section mild steel beam has an ultimate strength 50% greater
than its yield strength (see Ex. 18.1), whereas for an I-section beam the margin is in the
