430 12 Plastic Behavior of Structures
12.3 Fundamental Methods of Plastic Analysis............................
Analysis of the plastic behavior of a structure may be performed also by kinematical
and static methods. Both these methods are exact and much easier than the direct
method. The first of these methods deals with variousformsof failure, while the sec-
ond method deals with variousdistributionsof internal force satisfying equilibrium
conditions. The idea of these methods is explained below.
Absolutely rigid rod is suspended by four hangers 1–4 as shown in Fig.12.3a.
The axial stiffness of all membersEAis constant. Find the limit load considering
both methods.
12.3.1 Kinematical Method..........................................
This method requires consideration of differentforms of failureof a structure. For
each form of failure, there is a correspondingwell-defined value of the failure load.
The actual limit load is aminimumload among all possible failure loads.
Let us consider all possible forms of the failure of the structure; they are shown
in Fig.12.4a. Each form 1–4 presents position of the rigid rodMKfor different
scheme of failures. Assume that elastic displacements of hangers are much less then
plastic ones and may be ignored. Therefore, each position is obtained by rotation
of the rod around the point of connection of a hanger and the rod. Internal forces,
which arise in each hanger, areNy.
Now for each scheme of failure, we need to find the loadPusing equilibrium
condition.
Scheme 1:PDNy.dCd2dC3d /D6Ny
Scheme 2:PD1, since the moment arm of the forcePwith respect to point
of rotation is zero
Scheme 3:PDNy.2ddCdCd/D4Ny
Scheme 4:PDNy.3d2dC2dCd/D3Ny
Corresponding values ofPis shown on the schemes 1–4. The minimum value
ofPD3Ny. This case corresponds to rotation of the rod around the pointD.This
result coincides with result, which was already obtained by direct method.
12.3.2 Static Method.................................................
According to this method, it is necessary to find all possibledistributionsof internal
forces in statically indeterminate structure; we assume that for each distribution, the
internal forces do not exceedthe limit load. For each distribution of internal force,
there is a corresponding well-defined value of the external loadP. Actual limit load
is a maximum load among all possible limit loads.