In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Mathematics Based on Newton's laws of motion, the equilibrium equations available for a two-dimensional body are: the vectorial sum of the forces acting on the body equals zero. This translates to: the sum of the horizontal components of the forces equals zero; the sum of the vertical components of forces equals zero; the sum of the moments (about an arbitrary point) of all forces equals zero. In the beam construction on the right, the four unknown reactions are , , , and . The equilibrium equations are: Since there are four unknown forces (or variables) (, , , and ) but only three equilibrium equations, this system of simultaneous equations does not have a unique solution. The structure is therefore classified as statically indeterminate. To solve statically indeterminate systems (determine the various moment and force reactions within it), one considers the material properties and compatibility in deformations. Statically determinate If the support at is removed, the reaction cannot occur, and the system becomes statically determinate (or isostatic). Note that the system is completely constrained here. The system becomes an exact constraint kinematic coupling. The solution to the problem is: If, in addition, the support at is changed to a roller support, the number of reactions are reduced to three (without ), but the beam can now be moved horizontally; the system becomes unstable or partly constrained—a mechanism rather than a structure. In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it is preferable to use the phrase partly constrained here. In this case, the two unknowns and can be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields the same results as previously obtained. However, it is not possible to satisfy the horizontal force equation unless . Statical determinacy Descriptively, a statically determinate structure can be defined as a structure where, if it is possible to find internal actions in equilibrium with external loads, those internal actions are unique. The structure has no possible states of self-stress, i.e. internal forces in equilibrium with zero external loads are not possible. Statical indeterminacy, however, is the existence of a non-trivial (non-zero) solution to the