Advanced Methods of Structural Analysis

426 12 Plastic Behavior of Structures

Let us consider plane bending of a beam of a rectangular.bh/cross section.
In the elastic region of the stress–strain diagram, the normal stresses are distributed
within the height of a cross-section of the beam linearly. The maximum tensile and
compressed stresses are located at the extreme fibers of the beam. The stressy
corresponds to yield plateau (Fig.12.2a). Increasing of the load leads to appearance
and developing of the yield zone and decreasing of the “elastic core” of the section
of the beam. Diagrams in Fig.12.2b, c correspond to partially plastic bending of a
beam, which means that the middle part of the cross-section is in elastic condition,
while the bottom and top parts of the beam are in plastic condition. Further increas-
ing of load leads to complete plastic state (Fig.12.2d), which corresponds to the
limit equilibrium, i.e., we are talking about appearance of so-calledplastic hinge
(Fig.12.2d, e). It is obvious that all sections of the beam are in different states.
Defining of the location of the plastic hinge is an additional problem of plastic anal-
ysis. This problem will be considered below.

Plastic hinge

sy

sy

sy sy

F

F

h

Plastic zone

Plastic zone

Elastic

core

b

a

d e

b c

Fig. 12.2 Distribution of normal stresses within the height of a beam

What is the difference between plastic and ideal hinge? First, the plastic hinge
disappears if the structure is unloaded, so the plastic hinge may be considered
as fully recoverable or one-sided hinge. Second, in the ideal hinge, the bending
moment equals to zero, while plastic hingeis characterized by the appearance of
bending moment, which is equal to the limit (or plastic) moment of internal forces
FDybh 2 (Fig.12.2d). A bearing capability of a structure is characterized by the
plastic moment

MpDF

h

2

Dy

bh^2

4

:

Plastic analysis involves determination of plastic load or limit load, which struc-
ture can resist before full failure due to yielding of some elements. The limiting
load does not depend on settlements of supports, errors of fabrication, prestressed