# 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.

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.