Advanced Methods of Structural Analysis

436 12 Plastic Behavior of Structures

Assume that in the limit state, the vertical displacement at the each forcePequals
unity. Then the angle of inclination of theleft and right parts of each mechanism is

'D ̇

1

l=2

D ̇

2

l

:

The work done by limit loads equalsW.P /D2Plim 1 .TheworkW 1 produced by
limit momentsMyat the intermediate support equalsW 1 D2My'.TheworkW 2
produced by limit moments at the points of loadPequalsW 2 D4My'(these
plastic moments are shown according to location of extended fibers in elastic stage).
SinceW.P/CW 1 CW 2 D 0 , then this condition can be rewritten as follows:

2Plim 1 D2My'C4My'D

12My

l

:

The limit load becomes

PlimD

6My

l

:

As it was expected, results obtained by both static and kinematical methods are
identical.
Some typical examples of plastic analysisof statically indeterminate beams are
shown below.

Example 12.1.Design diagram of pinned-clamped beam, which is subjected to
concentrated loadP, is presented in Fig.12.6a. Calculate the limit loadP.

Fig. 12.6 Plastic analysis
and graphical calculation of
limit load for pinned-clamped
beam

AB

l/ 2 l/ 2

P

a

A B

P

b M

y

A B My

P

c

My

My/ 2

4 My

Pl

d

Solution.From the elastic analysis of the beam we know that the maximum mo-
ment occurs at the supportB. Therefore, just here will be located the first plastic