Advanced Methods of Structural Analysis

12.4 Limit Plastic Analysis of Continuous Beams 439

My

My

1

lim

l

P ab

8

qliml 22

LPM

LPM

c

K D

My P My q My

b

A

K B C D

P

a= 3 m

q

b= 4 m

l 1 =7m l 2 =6m l 3 =2m

a

P=2ql 2

Fig. 12.8(a) Design diagram; (b) Beam with plastic hinges at supports; (c) Bending moment
diagram inscribed between two limit plastic moments (LPM)

Solution.The given structure has two redundant constraints. There exist different
failure mechanisms. Let us consider one of them. The progressive increase of the
loads leads to the appearance of the plastic hinge at one of the supports, so the
structure becomes statically indeterminate of the first degree. Further increase of
the loads leads to the appearance of the plastic hinge at another support. Finally,
plastic hinge happens at the last support. It means that the entire continuous beam
is being transformed into two simply supported beams subjected to given loads and
plastic momentsMyat the supportsA, BandCasshowninFig.12.8b. Direction of
momentsMyis shown according to location of extended fibers in elastic analysis.
The order of appearance of the plastic hinges on supports depends on the relation-
ships between forcePand loadqas well as geometrical parameters of the beam.

This failure mechanism allows developing the theory of plastic analysis of con-
tinuous beams subjected to several loads. However, it does not mean that exactly
the above sequence of formation of plastic hinges will be realized. For example, if
loadqis small then plastic hinges can appear first of all at the supportsA, B,andat
pointK, and after that at supportCand in the second span. Real sequence of plastic
hinges may be defined only after determination of limit load as shown below.
Bending moment at the pointKof the first span caused by forcePas well as
plastic moments at supportsAandBequals

MKD

Pab

l 1



Myb

l 1



Mya

l 1

D

Pab

l 1

My: (a)