Solution Procedures for Nonlinear Problems
For nonlinear problems, the stiffness of the structure, the applied
loads, and/or boundary conditions can be affected by the induced displacements.
The equilibrium of the structure must be established for a deformed shape
which is unknown and must be guessed. At each equilibrium state along
the equilibrium path, the resulting set of simultaneous equations will
be nonlinear. Therefore, a direct solution will not be possible and an
iterative method will be required.
Several strategies have been devised to perform nonlinear analysis.
As opposed to linear problems, it is extremely difficult, if not impossible,
to implement one single strategy of general validity for all problems.
Very often, the particular problem at hand will force the analyst to try
different solution procedures or to select a certain procedure to succeed
in obtaining the correct solution (for example, "Snap-through"
buckling problems of frames and shells require deformation-controlled
loading strategies such as displacement and arc-length based controls
rather than Force-controlled loading).
For these reasons, it is imperative that a computer program used for
nonlinear analyses should possess several alternative algorithms for tackling
wide spectrum of nonlinear applications. Such techniques would lead to
increased flexibility and the analyst would have the ability to obtain
improved reliability and efficiency for the solution of a particular problem.
Concept of Time Curve
For nonlinear static analysis, the loads are applied in incremental
steps through the use of "time" curves. A time curve prescribes
how a load or a restraint changes during the solution steps. For nonlinear
dynamic analysis, and nonlinear static analysis with time-dependent material
properties (e.g., creep), "time" represents the real time associated
with the loads' application. Click here
for an illustration of time curves.
The choice of "time" step size depends on several factors
such as the level of nonlinearities of the problems and the solution procedure.
A computer program should be equipped with an adaptive automatic stepping
algorithm to facilitate the analysis and to reduce the solution cost.
The following topics are discussed: