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Steel Designers' Manual - 6th Edition (2003)
The number of redundancies has now been reduced by one (since M B = M p) while
the compatibility condition represented by Equation (10.7) no longer applies. Substitution
of q A = wL 3 /144EI and M B = M p in Equations (10.4) and (10.7) gives
M
M
A
C
2
5wL
M
= -
48 2
2 7wL
M
= -
96 4
p
p
Using these new values for M A and MC the bending-moment diagram (Fig.
10.10(b)) is obtained, and inspection of the diagram shows that the second plastic
hinge can be expected to occur at end A. Putting MA = Mp gives
w
M 3 p 48 14. 4Mp
= ¥ = M = 08 . M
2 2
2 5L
L
C p
while Equation (10.3) gives D C = MpL 2 /16EI. The beam is now statically determinate
with MA = MB = Mp, and eventually the third hinge forms at C when w reaches
16Mp/L 2 (see Fig. 10.10(c)) and DC = MpL 2 /12EI. The important point to note is that,
despite the difference in initial conditions, the failure pattern, the failure load and
the deflection at the point of failure (relative to the ends) are the same for the two
cases. The uniqueness of the plastic limit load, i.e. its independence of initial conditions
of internal stress or settlement of supports, is a general feature of plastic analysis.
Deflections at the point of collapse can however be affected, as indeed they are
in the second case just described, when considered relative to the original support
position A0B (Fig. 10.10(a)).
10.6 Plastic failure of propped cantilevers
The case of the propped cantilever under a uniformly distributed load, Fig. 10.11(a),
cannot be solved quite so simply, and both upper and lower bound methods
described in Chapter 9 have to be used. A possible equilibrium condition is shown
in Fig. 10.11(b) where the reactant line a1B has been arranged so that the coordinate
at the left-hand support is equal to the co-ordinate of the resultant moment
diagram at mid-span. At the right-hand support the condition of zero resultant
moment has to be satisfied. If the equal moments at A and C are regarded as plastic
hinge (Mp) values, the mechanism condition is satisfied. Hence Mp is an upper bound
(unsafe) value and should be denoted by Mu. Considering the geometry of the
moment diagram at mid-span,
2 lwL Mu = Cc1 + c1c= + Mu ( Mu = Mp)
8 2
Plastic failure of propped cantilevers 339
where l is the load factor at rigid plastic collapse (see Section 9.3.6)