Statically Indeterminate System

in structural mechanics, a geometrically invariant system of structures in which constraints (forces at supports, bars, and elsewhere) cannot be determined by the equations of statics alone but require the examination of these simultaneously with additional equations characterizing deformations in the system. The necessary and sufficient condition of a statically indeterminate system is the presence of extra, or redundant, constraints that may be removed without violating the geometric invariance of the system. The number of additional equations, equal to the number of extra constraints (or extra unknowns), is called the degree of static indeterminateness of the system.

Forces may arise in the elements of statically indeterminate systems (in contrast to statically determinate systems) that result from settling of the supports, effects of changes in temperature, contraction of materials, and imprecision of assembly or manufacture. The distribution of forces in these systems depends not only on the load but also on the ratio between the cross-sectional dimensions of the individual elements; furthermore, if these elements are made of various materials, the distribution of forces also depends on the ratio between the moduli of elasticity of the materials. In statically determinate systems, the loss of only one constraint leads to the breakdown of the entire structure; statically indeterminate systems retain their load-bearing capacity (and geometrical invariance) after the loss of one or even all the redundant constraints. In this respect, statically indeterminate systems are more reliable than statically determinate systems.

In the two principal methods for calculating statically indeterminate systems, either forces or displacements are taken as the starting (extra) unknowns. In a third method, one portion of the unknowns is selected as forces and the other portion as displacements. The major difficulty in calculating systems with a high degree of static indeterminateness lies in the need to construct and solve a system of equations with a large number of unknowns; the use of computers permits the full automation of the time-consuming calculating process.