In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 177

Q ult = 4 M p /L Q ult = 4 M p /L

Beam

Rigid Rigid

Elastic Elastic

∞

Elastic-plastic

Plastic
hinge

Curvature

Q ult

Q ult

Deflected
shape

(a) Real beam

(b) Rigid-plastic beam

Figure 5.25 Beam with full plastic moment M p at mid-span.

Q

L/2 L/2

(a) Beam

5

Real

Q

M p M y

1 Elastic

3 4

2

First yield

3

Elastic-plastic

2

Rigid-plastic

4

Fully plastic

1

5

Strain-hardened

0

(b) Bending moment

(c) Deflection

Figure 5.26 Behaviour of a simply supported beam.

infinite curvature at mid-span causes a finite change θ in the beam slope, and so
the deflected shape of the rigid-plastic beam closely approximates that of the real
beam, despite the very different curvature distributions. The successive stages in
thedevelopmentofthebendingmomentdiagramandthecentraldeflectionofthe
beam as the load is increased are summarised in Figure 5.26b and c.