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Steel Designers' Manual - 6th Edition (2003)
340 Beam analysis
whence
(b)
2
lwL
Mc
= = 0. 0833lwL
12
(a)
Fig. 10.11 Plastic failure of a propped cantilever
I.. H
By a closer inspection of the resultant moment diagram, it is readily shown that the
maximum resultant moment does not occur at the mid-span, but between C and D,
at 5L/12 from the propped support. The value of this moment is lwL2 /11.52 (=
0.0868lwL2 ), which is in excess of Mu. Note that this value is based on the moment
at the fixed end being lwL2 /12. By making the plastic moment of the beam equal
to this higher value, a bending moment diagram satisfying the equilibrium and
plastic moment condition is derived, i.e. a static solution M1 = 0.0868lwL2 .The
required Mp lies between these limits, M1 and Mu, i.e.
0.0833lwL2 £ 0.0868lwL2 To obtain the exact value of M p, the sagging hinge is positioned at an unknown distance
x from the right-hand support (Fig. 10.11(c)). Considering the total ordinate
of the free-moment diagram at this hinge position, then
x
L
whence
M
lwL lw
Mp + Mp
= x - x
2 2
p =
lwL
Ê L-xˆ x
2 Ë L+ x¯
2
2
(c)
(10.8)