Civil Engineering Reference
In-Depth Information
Figure 3.10
Plastic potential.
Figure 3.11
Vector of plastic straining for different loadings.
in contrast to elastic straining where the strains depend on the increments of stress as
given by Eq. (3.27). Figure 3.11 shows two different loadings B
A, both
of which cause failure at A. The plastic strains are the same for both loading paths; they
are governed by the gradient of the failure envelope at A and not by the loading path.
The behaviour of an ideal elastic-perfectly plastic material can be represented by
the behaviour of the simple model illustrated in Fig. 3.12(a). This consists of a soft
rubber block with a frictional sandpaper base and a rigid platen bonded to the top.
A constant normal force
F
n
and variable horizontal forces
F
x
and
F
y
are applied to the
platen. If the horizontal forces are less than required to cause frictional sliding of the
sandpaper over the table all deformations of the platen are due to elastic deformation of
the rubber block. Thus increments of force
→
A and C
→
x
e
in the direc-
tion of the force as shown in Fig. 3.12(b). If, however, there is frictional sliding then the
direction of plastic (irrecoverable) displacement
±
δ
F
x
cause displacements
±
δ
p
is in the direction of the resultant
δ
force
F
and is independent of the increment of load
δ
F
x
or
δ
F
y
, as shown in Fig. 3.12(c).
3.10 Combined elasto-plastic behaviour
With reference to Fig. 3.1, the stress-strain behaviour is elastic up to the yield point
and is perfectly plastic at the ultimate state. Between the first yield and failure there
are simultaneous elastic and plastic components of strain.
