Advanced Methods of Structural Analysis

12.4 Limit Plastic Analysis of Continuous Beams 435

In the limit state, the bending moment at the point of application of forceP

should be equal to the limit moment, i.e.,

Pl

4



My

2

DMy: (12.1)

This equation leads to the limit load

PlimD

6My

l

:

Limit load can be also found using graphical procedure, based on (12.1)using

the superposition of two bending moment diagrams; this procedure is presented in

Fig.12.5e–h.

These diagrams are caused by limit plastic moment at the plastic hinge 1 and

loadP. Plastic hinge at support 1 with plastic momentM 1 DMyis shown in

Fig.12.5e; corresponding bending moment diagram is shown in Fig.12.5f. Bending

moment diagram for two simply supported statically determinate beams with forces

PisshowninFig.12.5g; the final bending moment at the force point isPl 4 M 2 y.

If this ordinate is less than plastic momentMy, then the limit state at the point of

application of force does not occur.

Let us find such value ofPso thatPl 4 M 2 yDMy. This could be found by the

procedure of equalizing of the final bending moment at the point of application of

force and the limit moment at support 1 as shown in Fig.12.5h by bold lines. This

procedure leads to the value of limit loadPlimD6Mly.

12.4.2 Kinematical Method..........................................

This method is based on the following idea: in the limit state, the total work done

by unknown plastic loads and all plastic bending momentsMyis zero. This method

consists of the following steps:

1.Identify the location of the potential plastic hinges. These hinges may be located
at supports and at points of concentrated loads. In case of distributed load, plastic
hinge may appear at point of zero shear. Also, plastic hinges may occur at the
joints of the frame. Thus, we identify possible failure mechanisms.
2.Equilibrium equations should be written for each failure mechanism. In our
case, the failure mechanism is shown in Fig.12.5i. This failure mechanism is
as follows: plastic hinge first appears at the middle support and thus, turning the
original statically indeterminate beam into two statically determinate simply sup-
ported beams. With further increase of loadP, plastic hinges appear at the points
of application of forcesP. As the result, three hinges are located on one line
and thus, the system becomes instantaneously changeable, which corresponds to
limit state.