In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 204

of maximum moment for the previous mechanism), whence

11.6567 M p
L 2

< q ult < 11.6575 M p .

L 2

These are very close to the exact solution of
q ult = ( 6 + 4 √ 2 ) M p / L 2 ≈ 11.6569 M p / L 2
forwhichthehingeislocatedatadistance(2 − √ 2 ) L ≈ 0.5858 L fromthebuilt-in
support.

5.11.3.3 Graphical solution

For the propped cantilever of Figure 5.31, the parabolic free moment diagram
for a simply supported beam is shown in Figure 5.39c. All the possible collapse
mechanisms have a plastic hinge at the left-hand support and a frictionless hinge
at the right-hand support, so that the reactant moment diagram is triangular, as
shown in Figure 5.39d. A first attempt at combining this with the free moment
diagramsoastoproduceatrialcollapsemechanismisshowninFigure5.39b.For
this mechanism, the interior plastic hinge occurs at mid-span, so that

M p + M p / 2 = qL 2 / 8

which leads to

q = 12 M p / L 2

as before.

q

2

qL /8 (+)

R L R R

L

(a) Propped cantilever (c) Free moment diagram

L /2 L/ 2

M p /2

(-)

M p

M p

( - )

2

( + )

qL /8

M max

M p

M p

(b) T ri a l c ol l ap s e m o m e n t d ia gram

(d) Reactant moment diagram

Figure 5.39 Graphical analysis of the plastic collapse of a propped cantilever.