12.4 Limit Plastic Analysis of Continuous Beams 433
A plastic analysis of idealized rigid-plastic structures may be performed using
two principal methods, namely static and kinematical methods. Fundamental con-
dition for both the methods is that in the limit plastic state, the bending moments
at all plastic hinges are equal to yielding momentMy, which is a characteristic of
material and cross-section of the beam. For plastic analysis by both methods, it is
necessary to show a collapse mechanism first.
Let us consider application of static and kinematical methods for plastic analysis
of continuous beams. Two-span beam of constant cross-section subjected to two
equal forcesPis shown in Fig.12.5a. It is required to determine the limit load.
12.4.1 Static Method.................................................
First, let us consider the behavior of structure subjected to given load and the failure
mechanism.
First stage In the elastic condition, the bending moment diagram is presented in
Fig.12.5b. Increasing of the loads leads to theincreasing of bending moment ordi-
nates. Since maximum bending moment occurs at the support 1.M 1 D3P l=16/,
then a material of the beam begins to yield at this support. Spreading of the plastic
zone at the support 1 during the increasing loading is shown in Fig.12.2. Thus, the
first plastic hinge appears at this support, and the entire two-span continuous beam
is transformed into two simply supported beams. The maximum possible bending
moment at the support 1 will be equal to the limit bending momentMy.Sothe
moment at plastic hinge will beMy. Corresponding design diagram is presented in
Fig.12.5c.
Second stageEach of these simply supported beams is subjected to forceP
and plastic momentMyat support 1. Corresponding design diagram is shown in
Fig.12.5d (in fact, Fig.12.5cand12.5d are equivalent). Bending moment at the
point of application of force equals to
M
l
2
DPl
4My
2:Again, we will increase the loadsP. It is clear that the maximum moment occurs at
the point of application of loadP. The spreading of elastic zone of material at this
point is as shown in Fig.12.2. Finally, a new plastic hinge occurs within the span (in
this simplest case, this plastic hinge will be located at the point of application of the
load). As the result, three hinges will belocated on the on the each span of the beam;
they are – hinge at supportA, hinge under the point of application of the loadP,
and hinge at support 1. The nature of these hinges is different. Hinge at the pointA
is ideal one, which represents support, while two other hinges are plastic ones, and
they are the result of exhausted bearing capability of the beam. The same situation
is with the second beam 1-B. Even though the hinges are of different nature, but
since they are located on one line, this leads to the failure of the structure.