In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 181

moment distribution defined by the assumed plastic hinges must be in static
equilibrium with the applied loads and reactions. This condition applies to all
beams, elastic or plastic. The mechanism condition is that there must be a suffi-
cientnumberofplasticandfrictionlesshingesforthebeamtoformamechanism.
This condition is usually satisfied directly by the choice of hinges. The plastic-
ity condition is that the full plastic moment of every cross-section must not be
exceeded, so that

− M p ≤ M ≤ M p . (5.35)

If the assumed plastic hinges satisfy these three conditions, they are the correct
ones, and they define the collapse mechanism of the beam, so that

! #

Equilibrium
Mechanism
Plasticity

" $

(Collapse mechanism) satisfies (5.36)

Theultimateloadcanthenbedetermineddirectlyfromtheequilibriumconditions.
However,itisusuallypossibletoassumemorethanoneseriesofplastichinges,
andsowhiletheassumedplastichingesmaysatisfytheequilibriumandmechanism
conditions, the plasticity condition (equation 5.35) may be violated. In this case,
theloadcalculatedfromtheequilibriumconditionisgreaterthanthetrueultimate
load( Q m > Q ult ).Thisformsthebasisofthemechanismmethodofplasticanalysis
Mechanism method
Q m ≥ Q ult

Equilibrium
Mechanism

satisfies . (5.37)

which provides an upper-bound solution for the true ultimate load.
A lower bound solution ( Q s ≤ Q ult ) for the true ultimate load can be obtained
by reducing the loads and bending moments obtained by the mechanism method
proportionally (which ensures that the equilibrium condition remains satisfied)
until the plasticity condition is satisfied everywhere. These reductions decrease
thenumberofplastichinges,andsothemechanismconditionisnotsatisfied.This
is the statical method of plastic analysis
Statical Method
Q s ≤ Q ult

Equilibrium
Plasticity

satisfies (5.38)

If the upper and lower bound solutions obtained by the mechanism and statical
methodscoincideoraresufficientlyclose,thebeammaybedesignedimmediately.
If, however, these bounds are not precise enough, the original series of hinges
must be modified (and the bending moments determined in the statical analysis
will provide some indication of how to do this), and the analysis repeated.
The use of the methods of plastic analysis is demonstrated in Sections 5.11.2
and5.11.3fortheexamplesofthebuilt-inbeamandtheproppedcantilevershown