In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 179

mechanism forms may be significantly higher than that at which the first hinge is
developed. Thus, a first hinge design based on an elastic analysis of the bending
moment may significantly underestimate the ultimate strength.
TheredistributionofbendingmomentisillustratedinFigure5.28forabuilt-in
beam with a concentrated load at a third point. This beam has two redundan-
cies, and so three plastic hinges must form before it can be reduced to a collapse
mechanism. According to the rigid-plastic assumption, all the bending moments
remain proportional to the load until the first hinge forms at the left-hand sup-
portAat Q = 6.75 M p / L . As the load increases further, the moment at this hinge
remains constant at M p , while the moments at the load point and the right-hand
supportincreaseuntilthesecondhingeformsattheloadpointBat Q = 8.68 M p / L .
The moment at this hinge then remains constant at M p while the moment at the
right-handsupportCincreasesuntilthethirdandfinalhingeformsatthispointat
Q = 9.00 M p / L .At this load the beam becomes a mechanism, and so the ultimate
loadis Q ult = 9.00 M p / L whichis33%higherthanthefirsthingeloadof6.75 M p / L .
The redistribution of bending moment is shown in Figure 5.28b and d, while the
deflectionoftheloadpoint(derivedfromanelastic-plasticassumption)isshown
in Figure 5.28c.

5.5.4 Plastic collapse mechanisms

Anumber of examples of plastic collapse mechanisms in cantilevers and single-
andmulti-spanbeamsisshowninFigure5.29.Cantileversandoverhangingbeams

QL QL

----- p
9.0

----- p

M C

9.0

C

Q

8.68 8.68

B

6.75 6.75

A

A B C

M B

Hinges

M A

M
--- p

L/ 3 2 L/ 3

B

1.0

( b ) Moment redistribution

( a ) B u i l t - i n b e a m (c) Deflection

Q Q Q

M p

M p

M p

M p

QL = 6.75 QL QL = 9.0

M p

= 8.68

M p

M p M p M p

M p M p

(d ) Ben d ing moment diagrams

Figure 5.28 Moment redistribution in a built-in beam.