Advanced Methods of Structural Analysis

424 12 Plastic Behavior of Structures

e

s

Proportional limit

Elastic limit

Yield stress

Yield plateau

Ultimate stress

Fracture stress

True

fracture stress

∗

∗

∗

∗ ∗

∗

Elastic

region

Plastic behaviour

O

Unloading line

Elastic

behavior

Yielding Strain

hardening

Necking

C D

a

e

s

∗

∗

∗

∗∗

A

Unloading path

Residual strain Elastic strain

Total strain

b

O M N

Fig. 12.1 (a) Typical stress–strain diagram for structural steel. (b) Loading-unloading diagram

Plastic behavior starts at the elastic limit. The regionCDis referred as theperfect
plasticzone. In this region, the specimen continues to elongate without any increase
in stress. Above the yield plateau, starting from pointD, the behavior of the speci-
men is described by nonlinear relationships". If the specimen will be unloaded
at pointA(Fig.12.1b), then unloading line will be parallel to the load straight line,
so the specimen returns only partially to its original length. Total strain of the spec-
imen isON, while the strainMNhas been recovered elastically and the strainOM
remains as residual one.
If the material remains within the elastic region, it can be loaded, unloaded, and
loaded again without significantly changing the behavior. However, when the load is
reapplied in a plastic region, the internal structure of material is altered, its properties
change, and the material obeys to Hook’s law within the straight lineMA; it means
that the proportional limit of the material has been increased. This process is referred
to as thestrain-hardening.
For plastic analysis, we change the typical diagram by its idealized diagram. Dif-
ferent idealized diagrams are considered inengineering practice. Some of idealized
models are presented in Table12.1.