## P

Limit point

\

N Postbuckling point

N Postbuckling point

### FIGURE 24.27 Instability points.

the stiffness is lost completely. On the other hand, a structure, such as a dome, may lose its stability by a sudden buckling into a mode of deformation before the limit point, which occurs at a distinct critical point — the ''bifurcation point'' on the load path. It should be noted that the initial imperfection of the structure will greatly reduce the value of critical load, and certain types of space frame are very sensitive to the presence of imperfection.

In the stability analysis, usually the characteristic at a certain special state is investigated, that is, the stability mode and critical load are analyzed as an eigenvalue problem. The researchers are now more interested to study the whole process of nonlinear stability. As a result of the development of the computer matrix method, numerical analyses of the large system have become straightforward. Therefore, the discrete analysis of a space frame, itself a discrete structure, is very suitable for the study of stability problems. Major problems encountered in the nonlinear stability process are mathematical and mechanical modeling of the structure, numerical technique for solving nonlinear equations, and tracing method for the nonlinear equilibrium path. A great many research works have been carried out in the above area.

The Newton-Raphson method or the modified Newton-Raphson method is the fundamental method for solving the nonlinear equilibrium equations and has proved to be one of the most effective methods. The purpose of tracing the nonlinear equilibrium path is as follows: (a) to provide equilibrium analysis for the prebuckling state, (b) to determine the critical point, such as the limit point or bifurcation point, on the load path and its critical load, and (c) to trace the postbuckling response. On the basis of the increment-iteration process for finite element method, techniques for the analysis of nonlinear equilibrium path and its tracing tactics have made significant progress in recent years. Numerical methods used for the construction of equilibrium path associated with nonlinear problems, like load incremental method, constant arc-length method, displacement control method, etc., were developed by different authors. Since each technique has its advantages and disadvantages in the derivation of fundamental equations, accuracy of solution, computing time, etc., the selection of an appropriate method has a profound influence on the efficiency of computation. In the present stage of development, complicated equilibrium path can be traced with the aid of the above technique. Computer programs have been developed for the whole process of nonlinear stability and can be used for the design of various types of latticed shells.

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