432 12 Plastic Behavior of Structures
The cases 3 and 4 should be omitted, because internal forces are greater thanNy.
Now we need to consider cases 1 and 2 in more detail.
Case 1: SinceN 4 D 0 , then equilibrium equationP
YD 0 leads toPD3Ny.
Case 2: SinceN 3 DNy, then the same equilibrium equation leads to result:
PD3NyNyD2Ny.The maximum load which the system can resist is3Ny; thus, the actual limit
plastic load corresponds to case 1. This result have been obtained earlier by direct
and kinematical methods.
Summary For any statically indeterminate structure, there are a number of various
forms of failure, which are possible. Thekinematical theoremstates that the true
form of collapse is that one that corresponds to the minimum value of the limit load.
For any statically indeterminate structure, there are a number ofinternal force
distributionssatisfying equilibrium conditions. Thestatic theoremstates that the
distribution of internal forces, which occurs at the maximum value of the limit load,
corresponds to exhausted bearing capacity of a structure.
Both the methods express two extreme properties of plastic load for statically
indeterminate structure, all members of which obey Prandtl diagram.
12.4 Limit Plastic Analysis of Continuous Beams.........................
So far we have discussed plastic analysis of a structure in case of ideal elasto-
plastic material. Now we will consider the ideal rigid-plastic material (column 2,
Ta b l e12.1). When this stress–strain diagram may be adopted? Since elastic dis-
placements of a structure are significantly less than plastic displacements, then these
elastic displacements may be ignored. In this case, the material of the structure is
called as idealized rigid-plastic material, and the structure is called as rigid-plastic
one. This material is not real, but using this material, the procedure for plastic anal-
ysis of elasto-plastic structures may be simplified. This simplification is based on
the following fact: if two structures, which are made from elasto-plastic and rigid-
plastic materials, have the same limit plastic load, then the limit condition of the
elasto-plastic structure asymptotically approaches the limit condition of the rigid-
plastic structure.
Let a structure is subjected to different loads simultaneously. Assume that each
load may increase independently of each other. In this case, the limit conditions
may be approached under different values of loads. What will be the limit load in
this case? Concept of “limitload” becomes unclear. Therefore, let us assume that
the loading is simple. It means that if the structure is subjected to different loadsP 1
andP 2 DP 1 acting simultaneously, then these loads are increasing together and
parameterof the load (coefficient between loads) remains constant during entire
loading process.