Advanced Methods of Structural Analysis

442 12 Plastic Behavior of Structures

B 1

B 2 B 3

S

J

F B^3 +S+J

Design diagram

Fig. 12.9 Design diagram of the frame and possible mechanisms of failure

12.5.1 Beam Failure..................................................

This scheme is characterized by failure of its horizontal element only; the horizontal
displacement of the frame is absent (Fig.12.10b). In this case, the plastic hinges
appear at the jointsC(andD) as well as at the pointK. Corresponding plastic
moments are denoted asMyvertandMyhor. Their directions are shown according to
the location of extended fibers; location of such fibers are easy to find if bending
moment diagram is plotted in elastic state. The inclination of the partsCKandKD
is denoted by ̨.
Equation of limit equilibrium for beam scheme of failure can be found using
kinematical method (in the limit condition the total work done by unknown plastic
loads and all plastic bending momentsMyequals to zero):

Qlim

̨l

2

2Myhor ̨2Myvert ̨D0:

We can see that beam mechanism is realized only by loadQ.SinceMyhorD2Myvert,

then the solution of this equation leads to the following limit loadQlimD
12Myvert
l.
SinceQD2P, then corresponding limit loadPis following:

PlimD

6Myvert

l

: (12.2)