12.1 Idealized Stress–Strain Diagrams 425
Table 12.1 Idealized"diagrams for axially loaded members
Material
Elasto-plastic with Rigid-plastic with
Elasto-plastic Rigid-plastic linear hardening linear hardening
sesya aAessysesya aa 0sesya 0For further analysis, we will consider idealized elasto-plastic material and rigid-
plastic material. We start from elasto-plastic material; corresponding diagram is
called Prandtl diagram. This diagram has two portions – linear “stress–strain” part
and the yield plateau. Elastic properties of material hold up to yield point stress
y. The yield plateau shows that displacement of material can become indetermi-
nately large under the same stress. Idealized elasto-plastic material does not have
the effect of hardening. This diagram maybe applicable for a structural steel and
for reinforced concrete. Structural analysis on the basis of idealized diagram is re-
ferred as theplastic analysis. The quantitative results of plastic analysis are much
closer to the actual behavior of a structure than the results obtained on the basis of
elastic properties of material.
In case of statically determinate structure, yielding of any member leads to the
failure of the structure as a whole. Other situation occurs in case of statically indeter-
minate structure. Assume that for all members of the structure, the Prandtl diagram
is applicable. In the first stage, when loads are small, behavior of all members fol-
lows the first portion of the Prandtl diagram. Proportional increase of all loads leads
to the yielding in the most loaded member. It means that the degree of statical in-
determinacy is decreased by one. The following proportional increase of all loads
leads to the following effect: the internal force in theyieldingmember remains the
same, while the forces in theothermembers will be increased. This effect will be
continued until the next member starts to yield. Finally, the structure becomes stati-
cally determinate and yielding of any member of this structure immediately leads to
the failure of the structure, since the structure is transformed into a mechanism. In
general, if the structure hasnredundant constrains, then its failure occurs when the
number of yielding member becomesnC 1. Its means that capability of a structure
to carry out the increasing load has been exhausted. This condition is calledlimit
equilibrium condition. In this condition, the limit loads and internal forces satisfy to
equilibrium condition. The following increase of a load is impossible. In this condi-
tion, the displacement of the structure becomes undefined. While the linear portion
of typical stress–strain diagram leads to linear problems of structural analysis (elas-
tic problems), the Prandtl diagram leads to nonlinear problems of plastic behavior
of structures. Indeed, the design diagram of a structure is changed upon different
levels of loads. Transition from one design diagram to another happens abruptly.